Pore network tortuosity controls fast charging in supercapacitors

Abstract

Ionic transport within porous carbon electrodes is crucial for optimizing charge and discharge rates in supercapacitors, yet the material properties governing ion dynamics remain poorly understood. Unlike the traditional viewpoint, here we find that mesoporosity does not necessarily correlate with a high supercapacitor rate capability. We employed pulsed-field-gradient nuclear magnetic resonance to directly measure the anionic effective diffusivities in the carbon pores. This technique probes ionic transport in supercapacitors. Our findings reveal a major discrepancy between short-range and long-range diffusivities, which captures the tortuosity of the pore network. Short-range diffusivities lack correlation with supercapacitor rate capability, whereas long-range diffusivities correlate strongly. Low-tortuosity nanoporous carbon exhibited superior rate capability, which highlights the importance of well-interconnected pore networks for efficient ion transport. Our study reveals that the pore network tortuosity is a key factor governing charging rates in amorphous nanoporous carbon and that it can be used to guide the design of electrodes with optimized transport channels to enhance supercapacitor performance.

Supercapacitors are fast-charging energy-storage devices. However, an understanding of how structure impacts high-power energy storage is still lacking. Here pulsed-field-gradient nuclear magnetic resonance measurements show that the pore network tortuosity, rather than traditional porosity analyses, in porous carbon dictates the speed of supercapacitor charging.

Main

Electrochemical double-layer capacitors (EDLCs) are high-performance energy-storage devices known for their rapid charge and discharge capabilities and long cycle lives. These devices typically consist of nanoporous electrodes immersed in electrolyte solutions^1,2, with carbon-based materials being the most widely used due to their high surface area and chemical stability^1,3. This nanoporous carbon is composed of defective graphene-like fragments that form the pore walls of a three-dimensional porous network with a range of pore sizes⁴.

Although much of the current research is focused on increasing the energy density of EDLCs, either by maximizing the cell voltage or by improving the specific capacitance of the electrodes⁵, the role of electrode material and electrolyte in the rate capability, an advantage of EDLCs, is sometimes overlooked. Efficient ionic transport is crucial for maintaining performance at high charge and discharge rates^6–8, particularly in thick electrodes that can be used to increase the supercapacitor energy density^9–11.

Electrode mesoporosity (having pores with diameters of 2–50 nm) is often mentioned as a critical component for efficient ionic transport. The mesopores serve as a bridge between micropores (where charge storage occurs) and the bulk electrolyte^12–17. However, the relation between mesoporosity, ionic transport and performance remains insufficiently understood. Although some studies have demonstrated improved performance in materials with engineered mesoporous architectures (such as elongated channels designed to facilitate rapid ion transport^8–11,18 or electrodes with etched microporous patterns^19,20), others indicate that mesopores do not necessarily enhance ionic transport in carbon materials^21,22. Disentangling the relative contributions of pore architecture, meso- and macroporous specific surface area (SSA), and ionic diffusion to supercapacitor performance remains a substantial challenge, and existing studies have been, in part, limited by the lack of characterization of ionic transport in such complex porous structures, particularly in commercial nanoporous carbon.

Several techniques have been employed in attempts to analyse the pore structures of nanoporous carbon materials, each with inherent limitations. Gas physisorption, the standard method for evaluating the pore size distribution, relies on small gas molecules, so that it can overestimate the surface area and transport pathways accessible to larger electrolyte ions²³. A pair distribution function analysis is effective for probing short-range atomic structures but is less well suited for investigating long-range pore connectivity or network organization²⁴. Crystallographic techniques, like X-ray diffraction, are similarly limited by the lack of long-range order in amorphous carbon, whereas advanced tomographic methods, such as focused-ion-beam nanotomography, are restricted in resolution and cannot resolve micropores (<2 nm)^25,26.

One approach for relating the porosity of a material to its ability to transport ions or charge is the Bruggeman equation, which has been widely used to link porosity to transport properties in materials such as the graphite electrodes of a lithium-ion battery^27,28. However, its assumptions of uniform particle distribution and isotropic pores are inadequate for nanoporous carbon, which features irregular shapes, broad size distributions and highly disordered, hollow structures. Consequently, studies investigating the role of mesoporosity in ionic transport often rely on transmission electron microscopy, which tends to emphasize and more readily characterize ordered regions of the structure, potentially overlooking the transport behaviour in disordered regions of amorphous carbon^12,14,16. These limitations underscore the need for experimental approaches that directly probe electrolyte transport within the pore network. Such approaches are critical for understanding the role of mesoporosity in facilitating ion movement and optimizing supercapacitor performance at high current densities.

Diffusion pulsed-field gradient (PFG) nuclear magnetic resonance (NMR) is a powerful technique for measuring transport in porous media and for investigating pore structure^29,30. PFG NMR can measure the self-diffusion coefficients of molecules confined in the porosity, and it can characterize diffusion restrictions due to pore size, interconnectivity or surface interactions^31,32. Tortuosity measurements obtained from PFG NMR have proven to be valuable for probing transport phenomena, and PFG NMR has been widely applied to the transport of hydrocarbon and shale gas in porous rocks^33,34 and transport in ion-exchange membranes^35,36. Tortuosity characterizes transport within a material. It describes the winding and indirect pathways for diffusion by highlighting the presence of dead ends, pore interconnectivity and multiscale diffusion properties^35,36. In carbon-based EDLCs, in situ PFG NMR has been used to show that the ion diffusion coefficients are substantially reduced in nanoporous electrodes compared with the bulk, as the coefficients are modulated by changes in the ion population at the electrode–electrolyte interface during charging. Thus, PFG NMR has provided insights into the charging mechanism³⁷. Other studies have used PFG NMR to investigate electrolyte diffusion in the hierarchical structures of nanoporous carbon, highlighting the interplay between diffusion, microporosity, mesoporosity and hierarchical pore architectures^21,22. Although these template-assisted materials have provided initial insights into ion transport within complex structures, questions remain regarding diffusion across various length scales, pore connectivity and the characterization of pore architecture in more disordered, template-free materials, which are more commonly used for industrial production. Addressing these aspects is crucial for evaluating charge and discharge rates and large-scale ion transport.

In this study, we discover a connection between the fast-charging performance of supercapacitors and the ionic effective diffusion coefficient in the electrode nanopores. We assess the fast-charging capabilities of nanoporous carbon supercapacitors with electrochemistry experiments and employ PFG NMR to investigate the underlying mechanisms affecting performance. We hypothesize that increasing the tortuosity of the pore network impedes the transport of electrolyte ions to charge storage sites. By probing diffusion at >3 µm length scales (D_∞), our results reveal a positive correlation between effective long-range diffusion and the rapid charge and discharge behaviour of supercapacitors. This relation is primarily influenced by material tortuosity. Building on previous NMR studies of supercapacitors^37–39, our work identifies the pore network tortuosity as a critical factor linking electrode structure to charge and discharge rates, and it demonstrates how NMR can provide unique insights into the mechanisms driving supercapacitor efficiency.

Evaluating the role of mesopore surface area and ion transport in supercapacitor rates

Electrochemical measurements on a group of activated carbon cloths (ACCs) showed similar capacitances at a low current density (50 mA g⁻¹) in 1 M tetraethylammonium tetrafluoroborate (TEABF₄) in acetonitrile (ACN) (Fig. 1a), indicating that a similar amount of energy can be stored and released upon slow charging and discharging. However, the capacitance decreases with increasing current density at rates that can vary notably between the different carbon materials, highlighting the necessity of characterizing and reporting electrode capacitance at a range of current densities to fully capture the performance of an electrode material (Fig. 1a).

Fig. 1. Supercapacitor rate capability.

a, ACC series with different pore size distributions with 1 M TEABF₄ electrolyte in ACN. b, ACC-15 with 1 M TEABF₄ (ACN)_, 1 M TPABF₄ (ACN) and 1 M TBABF₄ (ACN) electrolytes with increasing cation size.

Source data

To quantify the rate performance of electrode materials, we fitted the capacitance versus current density data with a decaying exponential function to obtain the rate capability J₀, which represents the current density at which 63% of the initial capacitance is lost (Fig. 1). Higher values of J₀ indicate better supercapacitor rate performance. We found J₀ values ranging from 1.6 A g⁻¹ for ACC-10 up to 91.1 A g⁻¹ for ACC-20 electrodes in 1 M TEABF₄ electrolyte in ACN (Supplementary Fig. 1 and Supplementary Table 1). As ACC-20 has larger pores than ACC-10 (Supplementary Fig. 2), we hypothesized that this would reduce the restrictions on ionic transport during charging, leading to better rate performance. To test this hypothesis, we systematically varied the electrolyte cation size using a fixed electrode material (ACC-15), with TEABF₄, tetrapropylammonium tetrafluoroborate (TPABF₄) and tetrabutylammonium tetrafluoroborate (TBABF₄) electrolytes (1 M in ACN). Increasing the cation size reduced the fast-charging performance (Fig. 1b and Supplementary Table 1) owing to the more restricted porosity accessible to the electrolyte^40,41.

Mesoporosity (pore sizes >2 nm) is often associated with providing transport channels for rapid electrolyte movement, whereas microporosity (pore sizes <2 nm) is linked to enhancing performance at low current densities due to its charge-storing surface interactions^13,15,17,42. To examine the role of mesoporosity on rate capability, we performed rate capability measurements of nanoporous carbon with various combinations of micro- and mesopores (Supplementary Fig. 2) and analysed the correlations between rate capability and mesopore and micropore SSAs (Fig. 2a,b). We did not observe a strong correlation between rate capability with microporosity or mesoporosity measured for carbon and films (Fig. 2a,b). This indicates that mesoporosity is not the primary determinant of rate capability^21,22. SSA measurements obtained from N₂ adsorption can overestimate the surface area accessible to electrolyte molecules, as the narrow micropores accessible to N₂ molecules might be inaccessible to the larger electrolyte ions. As a result, the SSA derived from N₂ adsorption may not accurately represent the active surface area involved in electrochemical processes. Importantly, supercapacitor performance is not driven solely by surface area but by the efficient transport of ions through the pore network^21,22, which is expected to depend on pore interconnectivity. Therefore, we hypothesized that directly studying electrolyte transport over longer length scales within the porous network using PFG NMR would provide a more accurate depiction of the factors determining good performance at high current densities.

Fig. 2. Correlation between rate capability and porosity metrics from gas sorption.

a,b, Correlations between the SSAs of micropores (<2 nm) (a) and mesopores (≥2 nm) (b) and electrode rate capability obtained from N₂ isotherms at 77 K using a slit-pore carbon model and quenched-solid density functional theory. Error bars represent the 95% confidence intervals of the rate capability fits.

Source data

PFG NMR used to measure diffusion and tortuosity in nanoporous carbon

PFG NMR probes ionic effective diffusion and tortuosity in the carbon nanopore network. It is a relevant approach because it characterizes how electrolyte ions navigate the meso- and microporous structure of the electrode under conditions like those in actual supercapacitor operation. To measure diffusion within the porosity of nanoporous carbon, here defined as the effective diffusion coefficient over a timescale (and, therefore, length scale) set by the experimental parameters, nanoporous carbon was loaded into an NMR tube, which was then filled with an excess of electrolyte to saturate the carbon pores in the absence of any external potential (Fig. 3a). In the ¹⁹F NMR spectra of ACC electrode materials soaked in 1 M TEABF₄ electrolyte in ACN, we resolved in-pore and ex-pore anionic environments^43–45 (Fig. 3a). In this context, ex-pore environments are where anions occupy the interstitial spaces between nanoporous particles and in-pore environments are where ions are confined within the carbon nanopores. PFG NMR measurements use gradient pulses to encode and decode the position of nuclear spins. If the positions of nuclei change during the experiment, for example, because of self-diffusion, the signal measured at the end of the experiment decreases, allowing the diffusion coefficient to be determined⁴⁶.

Fig. 3. ¹⁹F PFG diffusion coefficient measurements of ACC-15 soaked in 1 M TEABF₄ in ACN.

a, ¹⁹F NMR spectrum recorded at 9.4 T (corresponding to 400 MHz ¹H Larmor frequency) revealing a free electrolyte environment (red, ex-pore) and an in-pore environment (green). b, Decay of the ¹⁹F signal intensity in the in-pore environment with an increase in the gradient strength revealing the presence of several components. c, Schematic of diffusion in nanoporous carbons, highlighting the ex-pore component (red), in-pore to ex-pore exchange (magenta) and slow component (blue). Compare the winding diffusion path (orange line) to transport in a straight line (green dashed line). d, Dependence of the diffusion coefficient for the in-pore environment with diffusion time (equation (2)). Inset: The same data replotted on a logarithmic scale. e, r.m.s. displacement characterizing the diffusion transport distance in the gradient direction. Error bars were calculated based on the 95% confidence intervals from the diffusion measurement fits and rate capability measurements.

Source data

Initially, ACC-15 saturated with 1 M TEABF₄ (ACN) was studied using an intermediate diffusion time (85 ms). Diffusion ¹⁹F PFG NMR measurements reveal two components in the in-pore anionic environment (Fig. 3b,c). The dominant slow-diffusing component (Fig. 3b,c, blue component) corresponds to ionic diffusion within the porosity, which is influenced by electrolyte/surface interactions, pore connectivity and geometry. This component reflects the intrinsic transport properties of the electrode material. By contrast, the faster-diffusing component (Fig. 3b,c, magenta component) is driven by in-pore to ex-pore anion exchange, that is anions leaving a particle to enter another one during the diffusion time, leading to a greatly increased effective diffusion coefficient. This is expected to vary with particle packing and particle size^22,37,47 but does not represent the intrinsic electrode properties and is, therefore, not analysed further. Unlike previous work³⁷, no external potential was applied, so the effective diffusion coefficients reflect the ease of ionic transport in the porosity without contributions from electrostatic interactions during charging and discharging.

For ACC-15 soaked with 1 M TEABF₄ (ACN), the effective anionic in-pore diffusion coefficient decreases with increasing diffusion time (Fig. 3d). This contrasts with the neat electrolyte (Supplementary Fig. 3), in which the diffusion coefficient stays constant. Here we consider in-pore diffusion in two limiting cases: (1) D₀, at a low diffusion time and (2) D_∞, at long diffusion times. Because of their finite strength, the PFGs cannot capture slow diffusion processes at very low diffusion times, so D₀ and D_∞ were obtained by fitting the data. The fitting equation was adapted from the Taylor expansion introduced by Mitra et al.⁴⁸, which is valid only in the limit of short diffusion times.By contrast, our generalized equation (equation (2)) satisfyingly captures the diffusion behaviour across both short and long diffusion times. The effective diffusion coefficient of anions at short diffusion time D₀ (1.5 × 10⁻¹⁰ m² s⁻¹; Fig. 3d) was found to be one order of magnitude below that of anions in bulk electrolyte (1.5 × 10⁻⁹ m² s⁻¹; Supplementary Fig. 3). This is due to the anionic adsorption of anions and solvent molecules within the pores, which slows down anionic diffusion. Over long diffusion times, ionic transport occurs across a larger pore network, which is measured by the effective long-range diffusion coefficient D_∞ and is expected to be hindered by the size of the constrictions connecting the pores and the degree of interconnectivity between them, both of which determine whether the porosity is accessible to the electrolyte and contribute to the long-range diffusion coefficient at long diffusion times. Indeed, we found D_∞ =5.8 × 10⁻¹² m² s⁻¹ for BF₄⁻ ions diffusing in the pores of ACC-15, which is much lower than D₀ (1.5 × 10⁻¹⁰ m² s⁻¹).

The observed dependence of the effective diffusion coefficient on the diffusion times is a way to probe diffusion at different length scales⁴⁹. The root-mean-square (r.m.s.) displacement in the gradient direction $\sqrt{⟨z^2⟩}=\sqrt{2D\mathrm{\Delta }}$, where Δ is the diffusion time and D the diffusion coefficient, provides an estimate of the transportation distance in the gradient direction z (ref. ⁵⁰). In this study, we were able to probe diffusion at the ~0.2–3 µm length scale (Fig. 3e and Supplementary Fig. 4). For the diffusion measurements made in this work, contributions from in-pore to ex-pore anionic exchange have been filtered out, and therefore, the measurements are limited when the r.m.s. displacement exceeds the particle size at longer diffusion times. Furthermore, note that the anionic r.m.s. displacements from self-diffusion are expected to be much smaller than those attained during charging when an electric field gradient driving ionic transport is applied. Therefore, the scope of this work is to characterize material properties rather than ionic transport across the electrode during charging.

The discrepancy between the diffusion coefficient found at short diffusion times (D₀) and long diffusion times (D_∞) represents the diffusion coefficients in the two limiting cases of short-range diffusion (<0.2 µm) and long-range diffusion (>3 µm) (Fig. 3e). This discrepancy can be quantified using tortuosity values³⁴, which are here defined as τ = D₀/D_∞, where the diffusion coefficients at short (D₀) and long (D_∞) diffusion times are obtained from PFG NMR. Tortuosity describes the extent to which the pathways within a porous material deviate from a straight line. It reflects the structural complexity of the material and is commonly quantified by comparing the effective distance travelled by diffusing particles with the shortest possible distance between two points (Fig. 3c, bottom). Given the large tortuosity values measured in our work, it was essential to extrapolate the short-range diffusion coefficient to values not limited by the finite gradient strengths imposed by the instrumentation, which set a lower bound on the diffusion times suitable for accurate diffusion measurements.

Pore network tortuosity hinders fast-charging supercapacitor performance

To investigate the link between ionic transport and rate capability, we conducted PFG NMR measurements on various carbon powders and cloths saturated with 1 M TEABF₄ (ACN), which was chosen to exhibit resolved in-pore and ex-pore environments in ¹⁹F spectra. These measurements were critical for separating the dominant bulk electrolyte signal from the much weaker in-pore contribution (Supplementary Figs. 5–8). The selected carbon materials represent a broad range of rate capabilities (Supplementary Fig. 1), graphene-like domain sizes³⁹ and pore size distributions (Supplementary Fig. 2). Scanning electron microscopy images of powders and films (Supplementary Fig. 9) show that all samples contained particles (>3 µm) larger than the in-pore r.m.s. displacement.

Nanoporous carbon electrodes traditionally require polytetrafluoroethylene (PTFE) binder to transform loose powders into self-standing films. However, this binder can obstruct pores and diminish electrochemical performance, and the film fabrication process considerably reduces the experimental throughput. We compared PFGs measurements between powders and films immersed in electrolyte to quantify these effects. Films containing 5 wt% PTFE exhibited an approximately 1.5-fold higher tortuosity and a corresponding 1.5-fold decrease in long-range diffusivity compared with unbound powders, directly demonstrating the restrictive effect of PTFE on electrolyte transport pathways (Supplementary Fig. 10). Notably, these differences were modest relative to the variations in tortuosity and diffusivity across our different systems. Together, these findings indicate that, although film preparation can slightly reduce transport, powder-based measurements still provide valid insights into pore accessibility and transport phenomena. Using powders also streamlines sample handling and enables more rapid screening of carbon materials.

The short-range (<0.2 µm) effective diffusion coefficient D₀ did not show any clear correlation with the rate capability (Fig. 4a), SSA or capacitance (Supplementary Fig. 11). By contrast, the long-range effective diffusion (>3 µm) D_∞ showed a clear correlation with the measured rate capability (Fig. 4b). Carbon and electrolyte combinations with slower long-range anionic diffusivity D_∞ typically have a lower rate performance, which indicates that long-range in-pore ionic transport (for example, ionic transport through the electrode) is much more important than short-range transport (for example, ions moving to form the electric double layer) in determining supercapacitor charging rates. The discrepancy between long-range and short-range ionic transport is further illustrated by the anionic r.m.s. displacements along the gradient axis (Supplementary Fig. 4), which indicate that typically much longer-range anionic transport is possible in samples with lower tortuosity. r.m.s. displacements from self-diffusion are typically much smaller than those attained during charging when an electric field gradient is applied. Consequently, D_∞ does not grow linearly with the rate capability and plateaus at values above 100 A g⁻¹. This indicates that power performance is not always limited by the approximately 3-µm ionic transport range observed in our experiments and that larger-scale structural features may play a role in enhancing performance.

Fig. 4. Correlation between in-pore diffusion coefficients, pore network tortuosity and rate capability for different carbon materials and electrolytes.

a, Correlation between short-range diffusion coefficient and rate capability. b, Correlation between long-range diffusion coefficient and rate capability. c, Correlation between long-range diffusion coefficient and mesoporous SSA (>2 nm porosity). d, Correlation between pore network tortuosity and rate capability. The colour of the data points represents the carbon material, and the shape represents the electrolyte. Error bars were calculated based on the 95% confidence intervals from the fits.

Source data

Interestingly, $D_{\infty }$ does not correlate with the mesoporous SSA (Fig. 4c), which further supports the idea that other factors beyond SSA play a role in determining ionic transport and performance. These factors could include pore interconnectivity or the relative size of electrolyte molecules to the pore and pore constriction sizes. This result is consistent with previous reports of increased performance in samples containing large transport channels^19,51.

To evaluate the contribution of pore interconnectivity, we measured the tortuosity of our samples (Fig. 4d). For the carbon–electrolyte systems studied, we found pore network tortuosity values ranging from approximately 8 to 105 (Supplementary Table 1 and Supplementary Fig. 12), which highlights the diversity in the long-range electrolyte transport in the different samples. The particularly high tortuosity values in some samples demonstrate that ionic transport can be limiting in some electrode materials, as the diffusion path length was more than two orders of magnitude longer than diffusion in a straight line in some samples. Excitingly, we found a good correlation between the tortuosity parameter plotted against rate capability (Fig. 4d) and, overall, that higher tortuosity was associated with lower electrode performance at high current densities owing to the more indirect ionic transport pathways. The tortuosity values did not correlate with the microporous or mesoporous SSA (Supplementary Fig. 11), indicating that, in addition to the pore sizes, the pore arrangements and how pores are connected are important considerations for fast-charging performance. Both D_∞ and tortuosity plateau at rate capabilities above 100 A g⁻¹, indicating that other factors may also limit supercapacitor performance at high current densities.

To elucidate the mechanistic origin of the tortuosity–rate capability relation, we considered the ion transport processes underlying supercapacitor charging. Charge storage proceeds through ion migration to electrode surfaces, where counter-ion adsorption and co-ion desorption form electric double layers. Two mechanisms may limit this process: ohmic polarization, arising from resistive losses along tortuous pathways, and concentration polarization, which develops when ion transport cannot match the applied current, generating concentration gradients that restrict pore accessibility. As the measured cell resistance shows no correlation with tortuosity (Supplementary Fig. 14), ohmic losses are probably masked by other resistive contributions, indicating that concentration polarization dominates the rate limitation.

The relation between tortuosity and rate capability can be described using the Nernst–Planck equation for ion transport⁵² (Supplementary Information Section 1). Here electrolyte transport constraints manifest as a decline in galvanostatic charge–discharge-measured capacitance at high current densities, which isolates transport-limited pore accessibility rather than RC time constants that convolve resistive and capacitive effects^53,54. In the regime where anionic electromigration becomes rate-limiting, the critical current density J₀ marks the onset of capacitance loss and scales inversely with tortuosity (J₀ ∝ 1/$\tau$). Experimentally, a linear correlation between J₀ and 1/$\tau$ (Supplementary Fig. 15) confirms that tortuosity directly governs the transport-limited rate capability of porous electrodes.

Interestingly, increasing the electrolyte cation size leads to lower measured long-range effective anionic diffusion coefficients and tortuosities (navy points in Fig. 4b,d and Supplementary Table 1). These variations reflect how cation size and solvation shells affect the accessibility of porosity to electrolyte ions, markedly influencing their ability to enter pores and modulate the available volume for anionic transport. Because electroneutrality must be preserved, if a cation cannot access a given pore, anions are also excluded from that space to maintain the charge balance. This effect becomes more pronounced in closed or poorly connected pores. As a result, cation size indirectly impacts anion mobility by restricting the fraction of the pore space that is functionally accessible. It is important to note that the observed diffusion coefficients reflect ion motion across the entire porous network, which spans micrometre-scale distances, but they are strongly governed by nanoscale effects such as pore accessibility, crowding and local confinement. The tortuosity metric helps to disentangle these geometric and structural effects from purely molecular-scale mobility (for example, as predicted by the Stokes–Einstein relation), thus allowing us to focus on how the material architecture modulates transport. These findings highlight the value of anionic diffusion as a sensitive probe of supercapacitor rate performance and demonstrate that ¹⁹F PFG NMR remains an effective tool for characterizing transport and pore connectivity, even when the cation properties are varied.

Using PFG NMR, we have shown that long-range ionic transport determines the rate performance of nanoporous carbon electrodes. Tortuosity measurements at the micrometre scale revealed that well-connected pore networks enable faster ion access, directly enhancing high-rate charging and discharging. These results establish tortuosity as a key structural descriptor linking pore architecture to electrochemical behaviour and show that PFG NMR is a robust and quantitative tool for its characterization.

Our findings indicate that the design of electrode materials should go beyond maximizing charge storage by focusing on engineering the pore connectivity to minimize transport limitations. By combining low-tortuosity networks with materials that efficiently store charge, future supercapacitors and potentially other electrochemical energy-storage devices could achieve faster and more energy-efficient operation under demanding current densities.

More broadly, this framework bridges microscopic transport measurements and macroscopic device performance and offers a route to rational design principles for porous materials. Approaches that integrate structural control across several length scales, from atomic-level ordering to micrometre-scale connectivity, are probably most effective in overcoming diffusion bottlenecks. Such strategies could inform the development of next-generation electrodes that deliver both high energy and high power, thus accelerating progress towards sustainable, high-performance, energy-storage technologies.

Methods

Materials

YP50F and YP80F activated carbon powders were sourced from Kuraray, and PW-400 carbon powders came from Carbon Activated Corp. ACCs (ACC-5092-10, ACC-5092-15 and ACC-5092-20) were obtained from Kynol. PowerSorb EL-106 and EL-104 were obtained from Jacobi Carbons Group. TEABF₄, TPABF₄ and TBABF₄ salts (Acros Organics) were dried at 100 °C for 7 days and made into a 1 M solution with N₂-purged (3 h) anhydrous ACN from Sigma-Aldrich. Activated 3-Å molecular sieves were used to dry the electrolyte solution further.

NMR sample preparation

Nanoporous carbon powers were dried for at least 2 days in a vacuum oven at 100 °C and transferred to an N₂-filled glovebox where enough material was loaded in a 5-mm tube to fill the NMR coil. The carbon was covered with an excess of dry 1 M TEABF₄, TPABF₄ or TBABF₄ electrolyte in dry ACN (≥99.0%, Sigma-Aldrich). NMR tubes were sealed with PTFE tape, and the electrolyte was left overnight at room temperature to fill the sample porosity.

Coin cell assembly and electrochemical measurement

Electrochemical measurements were performed as in our previous studies^39,55. ACCs were directly used as electrodes, whereas powdered carbon was made into free-standing carbon electrode films by mixing carbon powder (95 wt%) with PTFE binder (5 wt%) in ethanol, rolling it to ~0.25 mm thickness and drying at 100 °C for 24 h to remove residual solvents. All electrochemical cells were symmetric two-electrode coin cells assembled in a N₂-filled glovebox. Electrodes were cut with a stainless-steel manual punch (6.4 mm diameter), while ensuring that the electrodes in each cell had the same mass within 0.2 mg, which ranged from 3.5 mg to 5.2 mg for carbon films and from 5.5 mg to 8.5 mg for ACCs due to their increased thickness. The mass loading ranged from 10.9 mg cm⁻² to 16.3 mg cm⁻² for films and 17.2 mg cm⁻² to 26.6 mg cm⁻² for ACCs. Electrodes were separated with a Whatman GF/A glass fibre separator, and approximately 150 µl of electrolyte was added. Stainless-steel (SS316) spacer discs and a spring were used as current collectors, and cells were sealed with a hydraulic crimper (Cambridge Energy Solutions) under a pressure of 80 kg cm⁻³ for approximately 40 s. All components were stainless steel and supplied by Cambridge Energy Solutions.

Electrochemical measurements were performed with a Biologic BCS-805 potentiostat. Constant current charge and discharge measurements were conducted for various current densities (from 0.05 A g⁻¹ to 2 A g⁻¹) with the same voltage window (0–2.5 V) for at least three cycles per condition. The capacitance was calculated from the slope of the second half of the discharge curve, with results averaged across at least two cells (standard deviation <5 F g⁻¹). Error bars reflect standard deviations across two cells. Sufficient cell precycling was verified by repeating charge and discharge tests at 50 mA g⁻¹ to ensure stable capacitance after pore wetting. The data are plotted in Supplementary Fig. 13.

To assess the rate capability across various nanoporous carbon materials and electrolytes, the capacitance C(J) was fitted against current density J using the expression $C\left(J\right)=C_0\exp (J/J_0)$, where J₀ represents the rate capability (Supplementary Fig. 1). To remove the contribution from the cell design, data points corresponding to less than 80% capacitance retention were excluded.

PFG NMR measurements

PFG NMR experiments were carried out using a Bruker Avance Neo spectrometer and a 5 mm H/F-BB diffusion probe at a magnetic field strength of 9.4 T (¹H Larmor frequency at 400 MHz) and 25 °C sample temperature. The diffusion experiments used stimulated echo and longitudinal eddy-current delay bipolar gradient pulses for diffusion and two spoil gradients that yielded high-resolution, distortion-free NMR spectra⁴⁶ (ledpbgp2s Topspin pulse sequence). The pulse durations of the diffusion gradients δ were each 1 ms long with pre-emphasis. The gradient strengths were varied from 17.1 mT m⁻¹ to 16.7 T m⁻¹ in 32 increments on an exponential scale. The gradients were calibrated using the self-diffusion coefficient of dry ethylene glycol⁵⁶ at 25 °C (8.87 × 10⁻¹¹ m² s⁻¹). Diffusion delays Δ ranged from approximately 3.5 ms up to 500 ms in 16 increments on a logarithmic scale. The sequence used two 8.5 mT m⁻¹ spoil pulses of 100 µs and 100-µs gradient recovery times. The optimized ¹⁹F 90° pulses on resonance was about 20 µs, which is approximately 12.5 kHz. The recycling delay was set to 5 s.

The in-pore environment was integrated in MATLAB, and the diffusion decay was fitted with the sum of two components⁴⁶:

$$I=\sum I_i\exp \left[-D_i{\left(\gamma g\delta \right)}^2\left(\Delta -\delta /3\right)\right],$$ 1

where D_i is the diffusion coefficient in m² s⁻¹, γ is the gyromagnetic ratio in rad s⁻¹ T⁻¹, δ is the summed duration of the two gradient pulses during the encoding or decoding, g is the strength in T m⁻¹ and Δ is the diffusion time in seconds.

The tortuosity $\tau$ and diffusion coefficient D₀ were obtained by fitting the diffusion coefficient D(Δ) versus the diffusion time, with the equation generalized from the Taylor equation in the limit of short diffusion times developed by Mitra et al.^31,48:

$$D\left(\Delta \right)=D_0\left[\left(1-1/\tau \right)\exp \left(-b\sqrt{D_0\Delta }\right)+1/\tau \right].$$ 2

Error bars show the 95% confidence interval.

Gas physisorption analysis

N₂ gas physisorption experiments were carried out using a high-vacuum physisorption and chemisorption analyser (Autosorb iQ from Anton Paar) at 77 K. For each carbon sample, around 25 mg of powder was placed into a glass bulb gas cell and degassed at 120 °C in vacuo for 16 h before the measurement⁵⁷. Pore size distributions were calculated with quenched-solid density functional theory (slit-pore model) based on a slit-pore model. The microporous and mesoporous SSAs were determined by integrating the pore size distribution plots for pores <20 Å and >20 Å, respectively.

Scanning electron microscopy

Scanning electron microscopy was performed using a Tescan CLARA 2 scanning electron microscope. Samples were imaged in ultrahigh resolution scan mode using an Everhart-Thornley secondary electron detector. The accelerating voltage was set to 5 keV, with a working distance of approximately 6 mm and a beam current of 61 pA. Uncoated samples were mounted on standard aluminium stubs using carbon tape and imaged under high vacuum.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41563-025-02404-6.

Author contributions

X.L. and T.K. prepared the samples. T.K. carried out the NMR measurements and performed the data analysis. X.L. conducted the electrochemical and gas sorption measurements and processed the raw data. T.K. developed the analysis scripts and analysed the datasets processed. T.K. wrote the paper with input from all authors. All authors contributed to the conceptualization and discussion of the results. A.C.F. provided guidance and acquired the funding.

Peer review

Peer review information

Nature Materials thanks Louis Madsen, Kyle Smith and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Data availability

The raw experimental data and processed data used in this study are available via the Cambridge Research Repository, Apollo, at 10.17863/CAM.115051 (ref. ⁵⁸). Source data are provided with this paper.

Code availability

The analysis scripts used in this study are available via the Cambridge Research Repository, Apollo, at 10.17863/CAM.115051 (ref. ⁵⁸).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41563-025-02404-6.