When hard work pays off

Abstract

We all know the adage: work smarter, not harder. Yet, there are times when making things harder—deliberately—can be unexpectedly rewarding. In this reflection, we revisit our study published ten years ago in the inaugural issue of Nanoscale Horizons, where we showed—through simulations and analytical theory—that introducing obstacles in charging microporous electrodes can paradoxically enhance energy storage and break the conventional trade-off between stored energy density and charging speed in electrical double-layer capacitors (S. Kondrat and A. A. Kornyshev, Nanoscale Horiz., 2016, 1, 45–52, https://doi.org/10.1039/C5NH00004A). Herein, we reflect on those original findings and present further examples of the broader principle: that under certain conditions, hindering charging can lead to greater energy storage in systems with low-dimensional (microporous) electrode materials.

A reflection on ‘Pressing a spring: what does it take to maximize the energy storage in nanoporous supercapacitors?’

Our paper (https://doi.org/10.1039/C5NH00004A),¹ published in the inaugural issue of Nanoscale Horizons, was accompanied by an inside cover cartoon—a simple image of a man pressing a spring (Fig. 1). The metaphor is almost trivial, yet deceptively deep: press a soft spring and it yields easily but stores little energy; press a stiff spring and, although it resists, the stored energy rises dramatically—provided the applied force is strong enough to compress it. Try to compress a spring that is too stiff with too little force, and the gain is close to nothing. Energy storage, then, is never free—it is a negotiation between effort and resistance, assuming an ideal, lossless system.

Fig. 1 The inside cover of the inaugural issue of Nanoscale Horizons. Can a simple spring analogy hold not only metaphorically but also functionally for modern energy storage systems? [The cartoon by Daria Kornysheva].

This principle is, of course, not new. One can find it in classical thermodynamic books, for example in Callen's treatment of generalized forces and displacements² and in Mazur's non-equilibrium formulations.³ In our paper,¹ we explored the nontrivial consequences of this simple concept and sought to give it teeth. What if this simple spring analogy holds not only metaphorically but also functionally for modern energy storage systems? What if resisting systems—supercapacitors with voltage-dependent capacitance—could actually benefit from their resistance to charging, i.e., their reduced ability to become charged under certain conditions?

In general, the energy stored in a supercapacitor is

where Q(u) is the charge stored in a capacitor at an applied cell voltage u, and C(u) = dQ/du is the voltage-dependent differential capacitance. This equation is often forgotten or neglected in favour of a simplified formula Cu²/2, which is, however, valid only in the linear regime, C(v) = const. In contrast to this commonly-used textbook expression, eqn (1) shows how a nonlinear, voltage-dependent capacitance can reshape the energy storage landscape. Specifically, low capacitance at low voltages can be tolerated if it enables higher capacitance at higher voltages—provided, of course, that such voltages can be applied. Our paper¹ explored how this simple insight plays out in nanostructured electrode materials, employing physical effects in which nontrivial capacitance–voltage dependence can be harnessed—sometimes counterintuitively—to yield more energy due to the “internal resistance” of a supercapacitor to become charged. In this note, we revisit the physical and technological implications of these findings, including their experimental realizations, and describe other examples illustrating how this generic concept can be leveraged to enhance energy storage.

The properties of ultranarrow pores

We begin by discussing the properties of ultranarrow pores, as found in microporous carbons and, more notably, in low-dimensional electrode materials such as graphene,^6,7 carbon nanotubes,⁸ and particularly MXenes.^9–11 A key feature that distinguishes such pores from wider (meso and macro) pores is their tendency to saturate with counter-ions at moderate to high—yet experimentally relevant—applied potential differences measured relative to the bulk electrolyte (Fig. 2a). Clearly, widening the pores can shift this saturation to higher potentials or help avoid it overall.

Fig. 2 Charge storage in ultranarrow pores. (a) Mesopores vs. low-dimensional pores (micropores). Unlike mesopores, low-dimensional pores saturate with counter-ions at moderate to high potential differences relative to the bulk electrolyte. (b) Capacitance as a function of potential (relative to the bulk electrolyte) for slit pores of different widths. Capacitance vanishes at high potentials because the pores become saturated with counter-ions. Simulation data from ref. 4. (c) Distribution of pore sizes of pitch-derived carbon (PC) and viscose-based carbon (VC) electrodes, and (d) their CV responses at a rate of 5 mV s⁻¹. Data from ref. 5.

Surprisingly, however, ultranarrow pores exhibit another distinct feature: the narrower the pore—down to the limit where ions can still enter—the higher its capacitance. This phenomenon, known as the anomalous capacitance increase, was first reported experimentally in ref. 12 and 13 and later attributed to the emergence of a superionic state in nanoscale conducting confinements.^4,14,15 Thus, increasing the pore width comes at a cost: the capacitance decreases, reducing the amount of stored energy. This trade-off is illustrated in Fig. 2b, which shows the differential capacitance obtained from Monte Carlo simulations of ionic liquid-filled slit nanopores of various widths.⁴ As the pore narrows, the capacitance increases, but the saturation voltage decreases, as indicated by the eventual vanishing of the capacitance at higher applied potentials. This behavior highlights the delicate balance among pore dimensions, applied potential, and energy storage efficiency.^16,17

These effects were investigated experimentally by Mysyk et al.,⁵ who studied two types of carbon electrodes: pitch-derived carbon (PC), characterized by a narrow pore size distribution (PSD) with a peak around 0.8 nm, and viscose-based carbon (VC), with a broader PSD (Fig. 2c). In both cases, cyclic voltammetry curves exhibited rectangular shapes, indicative of capacitive charge storage. However, when the cell voltage was increased to 3 V, the PC electrode showed a reduction in current—implying a decrease in capacitance—that can be attributed to micropore saturation. In contrast, the VC electrode displayed no such reduction at 3 V, maintaining a well-defined rectangular response (Fig. 2d).

Another distinct feature of ultranarrow pores is that they tend to become overcrowded with ions, resulting in sluggish charging dynamics.^18–20 Once again, widening the pores can accelerate charging^18,21—but, as discussed, this comes at the cost of reduced capacitance, underscoring a fundamental trade-off between energy and power density: increasing one typically comes at the expense of the other. Interestingly, it was later shown that sluggish charging can be accelerated—somewhat paradoxically—by deliberately slowing down the overall charging process.^20,22 The question we addressed in ref. 1 was, however, whether it is possible to overcome this power–energy trade-off altogether.

Ionophobic pores and the paradox of hindered charging

By investigating the charging dynamics of ultranarrow slit-shaped pores, we found that making pores free of ions at zero applied cell voltage— i.e., making them ionophobic—can drastically accelerate charging.^18,23 This is illustrated in Fig. 3a, which shows that an ionophobic pore charges more than ten times faster than a conventional ionophilic pore of the same width. The underlying reason is the enhanced, Knudsen-like diffusivity of counter-ions in initially empty pores, in contrast to ion-crowded pores where ion mobility is reduced and co-ions must diffuse out. Recently, Bo et al.²⁴ demonstrated such accelerated dynamics for MXene electrodes with increased OH termination, which created ionophobic conditions for sodium cations in the aqueous NaSO₄ electrolyte.

Fig. 3 Hard work pays off: ionophobic pores. (a) Ionophobic pores speed up the charging dynamics. Data from ref. 18. (b) Differential capacitance and (c) stored energy per surface area as a function of potential difference measured relative to the bulk electrolyte. The pore ionophobicity reduces the capacitance but shifts the counter-ion saturation to higher potentials, thereby enhancing the stored energy density. Data from ref. 1.

But what about charge and energy storage? Since ionophobic pores hinder ion entry, would they not also hinder charge storage? Our simulations, presented in ref. 1, showed that this is indeed the case—but only at low potential differences, u. As u increases, the accumulated charge gradually saturates to the same value as in ionophilic pores (unsurprisingly), and two additional effects take place: (i) the capacitance C is reduced compared to that of ionophilic pores (Fig. 3b). We showed that this reduction is because charging of ionophobic pores is governed essentially by counter-ion adsorption, which generally yields lower capacitance than counter-ion/co-ion swapping—the charging mechanism that significantly contributes to the charging of ionophilic pores.¹ (ii) More importantly, counter-ion saturation is shifted to higher potentials, as evidenced by the delayed vanishing of the capacitance in Fig. 3b. As a result, the stored energy—defined by eqn (1)—reflects a delicate interplay between the reduced capacitance and the delayed counter-ion saturation. Mathematically, because the capacitance C(v) is weighted by the potential v in the integrand of eqn (1), the overall effect of ionophobicity turned out to be beneficial for energy storage, as demonstrated in Fig. 3c.

Several groups have reproduced our predictions using different theoretical approaches and electrolytes. For example, Lian et al.,²⁵ despite questioning our conclusions in the title, arrived at the same findings using classical density functional theory. A few years later, Gan et al.²⁶ demonstrated the same effect for Emim-BF₄ ionic liquids via molecular dynamics simulations. Other groups have instead focused on developing methods to fabricate ionophobic pores. The first experimental realization was reported by He et al.,²⁷ who used in operando small-angle neutron scattering (SANS) to observe that sodium triflate (NaOTf) dissolved in dimethylformamide (DMF) remained outside the pores of a conductive metal–organic framework (MOF), with DMF preferentially occupying the pores; however, no electrochemical tests were performed. Yin et al.²⁸ proposed a fluorination treatment of microporous carbons with CF₄ plasma and observed a substantial increase in the wetting angle (from 19° to 72°) for tetraethylammonium tetrafluoroborate ionic liquid dissolved in propylene carbonate, along with improved electrochemical performance, particularly at high scan rates. Most recently, as already mentioned, Bo et al.²⁴ realized ionophobic pores for cations using MXene electrodes with increased OH termination.

Despite these encouraging findings, the technological implementation of pore ionophobicity remains an open question. Beyond the experimental and scalability challenges, an important issue is how ions redistribute within the EDLC device with ionophobic electrodes during charging. Paolini et al.²⁹ recently showed, through theoretical modeling, that ionophobicity may necessitate unusually wide membranes between the electrodes and induce substantial pressure variations inside the device. This latter effect could potentially compromise one of the principal advantages of EDLCs: their high cyclability.

Other routes to enhance energy storage by hindering charging

While wide-scale utilization of ionophobic pores remains challenging (yet, we believe, achievable^24,27,28), it has become clear that hindering charging can be realized through other means—yielding similarly beneficial consequences for energy storage. One such approach involves inducing an ordered ionic liquid structure inside the pores at zero applied cell voltage. Disrupting this ordered state requires applying an additional potential difference, thereby hindering charging and postponing nanopore saturation to higher voltages. This effect is illustrated in Fig. 4a, which shows the stored energy obtained from a lattice model for two types of pores:³⁰ one where an initial ionic ordering gives way to a voltage-induced phase transition into a homogeneous phase, and another where the ionic liquid remains homogeneous across the entire voltage range. Similarly to ionophobic pores, this order-induced hindered charging leads to a substantial increase in stored energy at intermediate and high voltages.³⁰

Fig. 4 Hard work pays off: other routes. (a) Enhancing energy storage by creating ordered ionic structures at zero polarization. Results of lattice modelling for slit nanopores, where a is the lattice constant, k_B is the Boltzmann constant, T is temperature, and e the proton charge. Voltage is measured in units of thermal voltage (k_BT/e ≈ 26 mV) and energy density is measured in units of k_BT per lattice cell area. The shaded area shows the region of the ordered ionic liquid phase in the case of low-voltage ordering. Data from ref. 30. (b) and (c) Enhancing energy storage using low quantum capacitance. (b) Capacitance and (c) stored energy for an ideally metallic nanotube and for metallic and semiconducting carbon nanotubes (CNTs) with chirality indices 7,7 and 10,2, respectively. The nanotube diameter is ca. 0.9 nm in all cases. Data from ref. 31.

Another mechanism involves the so-called quantum capacitance.³² Quantum capacitance, C_q, emerges due to a finite density of states (DOS) of electrons in electrode materials and contributes as a capacitance in series to the total capacitance of an electrode–electrolyte system:

C⁻¹(u) = C_q⁻¹(u_q) + C_EDL⁻¹(u − u_q), (2)

where C_EDL is the electrical double-layer capacitance and u_q is the gate voltage (i.e., the voltage drop across the electrode). In a metallic electrode, C_q is large compared to C_EDL and u_q≈0, so that C≈C_EDL. For non-ideal conductors such as graphene or graphite, C_q can be low and contribute significantly to the total capacitance. Long ago, Gerischer³³ demonstrated that C_q, rather than C_EDL, determines the total capacitance of an electrolyte–graphite system, substantially lowering the measured capacitance compared to that of the same electrolyte at gold electrodes. More recent studies^34–39 also reported reductions in the total capacitance—particularly at low voltages—due to low quantum capacitance, motivating efforts to enhance C_q (e.g., via nitrogen doping^40–42).

Thus, for low-DOS electrodes, the applied potential u is split between the electrode (u_q) and the electrolyte (u − u_q). Lower quantum capacitance requires a higher u_q—and thus a larger u—to achieve the same accumulated charge as in a metallic electrode. While this effect is generally detrimental for energy storage with flat electrodes and wide pores, it can be exploited to shift nanopore saturation to higher potentials. This is illustrated in Fig. 4b, which compares an ideally metallic nanotube (C_q = ∞, u_q = 0) with metallic and semiconducting carbon nanotubes (CNTs).³¹ While the CNT capacitance is low (and even vanishes for semiconducting CNTs) at low potentials, the saturation is indeed shifted to higher voltages, as indicated by the shift of the potential at which the capacitance vanishes (from ca. 0.9 V to 1.9 V). Similar to ordered ionic structures and ionophobic pores, such low-quantum-capacitance hindrance of charging can lead to markedly enhanced energy storage at intermediate and elevated voltages, as shown in Fig. 4c. As in ref. 31, this “less-is-more” concept has been proposed to apply to graphene-based electrodes,⁴³ which also exhibit a low quantum capacitance at low voltages.⁴⁴

Outlook

The central theme of this note—the principle of “hard work–high payoff” as applied to nanostructured, low-dimensional electrodes—continues to evolve from conceptual exploration toward practical engineering. We discussed three examples illustrating this principle: (i) pore ionophobicity, as originally proposed in our Nanoscale Horizons article¹ a decade ago (Fig. 3); (ii) ionic liquid ordering in non-polarized pores³⁰ and (iii) low quantum capacitance³¹ (Fig. 4). Since our original publication, ionophobic pores have been realized experimentally,^24,27,28 demonstrating enhanced electrochemical performance in line with our predictions.

We believe that low-dimensional materials such as graphene, CNTs, and MXenes—already increasingly pervasive across modern technologies—will enable further implementation of this principle for improved energy storage. In particular, the “less-is-more” concept rooted in low quantum capacitance holds promise for performance gains. With the possible exception of MXenes, many low-dimensional materials can be broadly classified as low-dimensional conductors (in addition to providing low-dimensional confinement for ions). In their undoped or non-polarized states, they typically possess a low density of free charge carriers; however, their electronic properties can be tuned via appropriate polarization strategies. Moreover, the thin pore walls of low-dimensional electrodes give rise to cross-wall interionic correlations,^45,46 making the pore wall thickness an additional parameter to manipulate the in-pore ion distribution and energy storage capabilities.⁴⁷ These tunabilities open up opportunities not only for advancing energy storage, but also for the development of entirely new device classes, including sensors, electroactuators, and desalination membranes.

Thus, while substantial challenges remain, the potential technological and functional benefits make this direction a compelling and worthwhile avenue for continued investigation.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the Polish National Science Centre (NCN) under grant no. 2021/40/Q/ST4/00160 to S. K.

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