Benchmarking the ragone behaviour and power performance trends of pseudocapacitive batteries

The ‘holy grail’ of energy storage is to achieve both high energy and high power densities ( ⩾100 Wh l−1 and ∼104 W l−1, respectively) as characterized in a Ragone plot. However, across the macroscopic dimensions over which energy storage systems operate, power performance is fundamentally limited by both drift and diffusion processes. In this work a macroscopic variation on the Gerischer–Hopfield formalism is applied to explore how the motion of electrical charges, moving between redox species, and screening counter-ions might be engineered in a pseudocapacitive system employing quantized capacitance (in the form of a pseudocapacitive battery) to reach this long-sought metric. Our theoretical findings show that the electron diffusion timescale between redox species generally determines power performance trends when pseudocapacitive coatings are applied monolithically. This electron-diffusion–dominated timescale, in turn, is shown to scale with the square of the coating thickness. However, when conducting pathways (or shunts) are introduced to substantially reduce the mean distance for electron diffusion the Ragone performance becomes dominated by ion drift and diffusion—even when the diffusion constants of all species are held equal. The resulting trends, for this shunting regime, show a power performance timescale that scales in a more linear fashion with increasing thickness of the redox-active region. By analyzing the Ragone performance metrics for realistic coating thicknesses between these two operational regimes, the resulting findings suggest that the diffusion constants needed to achieve the aforementioned high-performance metrics are plausibly achievable for both electronic and ionic charges in this proposed class of pseudocapacitive systems.


Introduction
Energy storage systems are an essential component in the global transition towards a sustainable energy paradigm [1].They are not only crucial to the operation of electric automobiles but also provide essential load-balancing support to renewable energy technologies that are hampered by intermittency-such as wind turbines and solar farms [2,3].However, energy storage systems must be carefully designed for each application to deliver the desired combination of energy and power performance [4,5].The Ragone plot, as depicted in figure 1(a), is often used to assess the degree to which these two capabilities can be satisfied by a given system [6][7][8].Energy density refers to a device's ability to store energy (horizontal axis in figure 1(a)), and power density refers to the speed at which a device can be charged or discharged (vertical axis in figure 1(a)).Many sustainable technologies such as electric vehicles are moving towards high-capacity and high-speed supercharging applications that are characterized by a combination of high energy and power density performance [9][10][11][12][13].Thus, a long-sought goal (or 'holy grail') of the energy storage research community has been to push the performance frontier further towards higher energy and higher power densities (i.e. the top right corner of figure 1(a)).However, the theoretical properties that impede or enable performance in this long-sought domain are not entirely understood [14][15][16][17].The two energy storage systems closest to reaching this ultimate goal are batteries and supercapacitors [18][19][20][21].Batteries are well-known for their high energy density, but their charging speed generally remains slow [22,23].By comparison, supercapacitors have superior charging speed but suffer from low energy density [24,25].Hence, the technological means by which both high power and energy density may be best achieved remains an open question.
Recently, it was proposed that one may dramatically enhance the energy density of supercapacitors through the use of quantized capacitance in the form of device that was termed a 'pseudocapacitive battery' (see the bronze region in figure 1(a)) [26,27].Such a pseudocapacitive battery consists of an electrolytic media sandwiched between two oppositely charged metal current collectors, where the surface is modified with a volume of deposited nanoparticles as illustrated in figure 1(b)-it is essentially a multiple redox-level variation on a redox-polymer battery [26].During the charging process, a potential bias on the electrode causes electron tunnelling and diffusion into the nanoparticles (through Faradaic means) to achieve energy storage via quantized capacitance.The ions in the electrolyte simultaneously migrate in response to screen the interfacial potential, leading to electric double layer (EDL) formation, as well as to charge balance Faradaic charges.Importantly, the energy density of pseudocapacitive batteries (or any such redox system) can be enhanced by: (1) developing an optimum active material to store a high density of electrons within the chosen electrolyte's stability window; and/or (2) increasing the loading of electrochemically active materials onto the electrodes to store more electrons (e.g.see figure 1(b)) [28].However, although the energy storage capability increases with the active material coating thickness, the power density is often impeded by thickening such a coating [28][29][30][31].Numerous discussions exist on the theoretical characteristics of the Ragone plot for supercapacitors and batteries, such that one may evaluate the chosen system's energy and power performance [17,32,33].However, the trade-off curvature between these two performance metrics for a quantized capacitance device has not been analysed.
The goal of this paper is to assess the power performance capabilities of systems utilizing quantized capacitance as an energy storage mechanism.This builds directly on the earlier findings in [26], wherein the potentially high energy density of such pseudocapacitive batteries were extensively explored and found to be plausible extendable to ⩾100 Wh l −1 .Our aim in this work is to assess whether realistically achievable diffusion constants can simultaneously deliver a targeted power performance of ∼ 10 4 W l −1 in figure 1(a).With the caveat that such performance must be achieved as practically relevant coating thickness approaches ∼50 µm [30,[34][35][36].Although the energy stored increases with active material coating thickness, the power density is often impeded by a thick coating [28][29][30][31].Thus, we wish to assess if energy systems employing quantized capacitance might feasibly deliver performance, under practically relevant dimensionalities, in the much sought-after domain of high energy and power density indicated in the upper right corner of figure 1(a).In the end, we shall show, the degree of feasibility is ultimately determined by the storage dimensionality and necessary diffusion constant.Our study towards this end is divided into several parts.In section 2, we first present our theoretical framework for exploring these properties across macroscopic length scales.Then in section 3, we present our analysis in two parts.First in section 3.1, we assess the performance properties in the regime where electron diffusion dominates (i.e.electron transfer (ET) between redox species in figure 1(b) [26]).Subsequently, in section 3.2, we examine the performance in the regime where electron and ion diffusion constants are matched in an optimized manner, both under the condition of monolithically applied coatings and when conductive pathways are introduced to accelerate electron transport.Our theoretical findings suggest that the targeted 'holy grail' combination of high energy and power performance (⩾100 Wh l −1 & 10 4 W l −1 ) is plausibly achievable in such pseudocapacitive batteries at practically relevant coating thickness and reasonably achievable diffusion constants.

Electron transfer and diffusion properties
Quantized capacitance (also known as 'solvated Coulomb blockade') relies on energy level quantization within nanoparticles to evenly space ET redox peaks by a constant charging energy over a wide potential range.This allows ET redox peaks to overlap and can theoretically provide a ideal near-rectangular voltammetry profile [27].Quantized capacitance theories based on the Gerischer-Hopfield formalism have been extensively discussed in previous publications [26,27].We shall not overview this formalism in detail here [37][38][39][40], but simply state that the effect of quantized capacitance (when U is sufficiently small) is to provide a continuum of redox levels with an energetic density of states (DOS) per electron-volt (eV) that can be reasonably approximated in volumetric units as where V d is the applied potential window, being split equally across both terminals, at which the maximum target concentration of electrons, ρ e,max , is stored [26] (see figures 1(b) and 2).
The assumption is that the potential drop at a given terminal (e.g.V d /2) is limited in magnitude only by electrolyte breakdown [26].
Here we denote ε to be the single-particle energy [41].The rate of interfacial electron transfer (k et ) is therefore determined by the interfacial potential drop (V int ) and the fraction of states in the reactant that are already filled as indicated in figure 2(a) [27,41].The substrate density of states D s and interface electron transfer coupling M et are both taken to be constant [41]-h is Planck's constant.Within a reasonable approximation, when ET peaks are overlapping sufficiently well, D e can also be treated as a constant per equation ( 1) [27].The potential (or relative local electrochemical potential inside the redox species) up to which electrons are already stored is determined by the concentration of electrons already stored pseudocapacitively at the interface (x = 0 in figure 2) which when included in equation ( 2) accounts for Pauli blocking, preventing the local placement of more than ρ e,max electrons at the chosen terminal potential V int ≈ V d /2.The number of stored electrons at any given time is given by ρ e .Moreover, the interfacial potential V int is a time dependent quantity due to the delayed onset of interfacial screening, as illustrated in figure 2(b) [42][43][44], which shall become important as we analyse the results later on.Interestingly, if we multiply equation ( 1) by the elementary charge (q = 1.6 × 10 −19 C) one ends up with a pseudocapacitive expression in terms of Farads per unit volume which is closely related to the concept of quantum capacitance that is typically applied in the physics literature [45].The only distinction is that equation (4) includes blockade interactions and thus can be, perhaps, more accurately referred to as quantized capacitance [46].When the charging energy U and/or potential drop splitting is not the same on both terminals, these equations have to be revised slightly but retain the same general physical argumentation [26].Now once an electron is transferred across the interface (see figure 2), it can diffuse between nanoparticles through an inter-particle hopping rate k ip , as indicated in figure 2(a).In this work we assume that the interfacial coupling (M et ) in equation ( 2) is sufficiently strong to make k et ≫ k ip .The electron hopping process governing k ip can also be captured through the Gerischer-Hopfield formalism as detailed in [26] and [27].Again, we shall not provide a detailed derivation here but simply point out that the inter-particle electron transfer rate can be directly translated into an electron diffusion constant in the form of where d is the system dimensionality and l ps is the mean distance between nanoparticles [26].This electron diffusion constant, and the underlying inter-particle electron transfer rate k ip , play a crucial role in determining the Ragone plot performance of a pseudocapacitive system.These diffusion aspects will be discussed in section 3. The alternative to employing an electron diffusion constant would be to track individual nanoparticles and their respective charge states, which is not feasible in the much larger continuum micron length-scale realm that must be explored to conduct a meaningful Ragone analysis per the goal summarized in figure 1(a).Within this context, experimentally, a typical pseudocapacitive coating thickness (e.g. on each terminal in figure 1(b)) is on the order of 50 µm or more [30,[34][35][36].

Electron and ion flow
The overall power density and volumetric charge storage of such a medium contain contributions from both electron and ion flow (see figure 2).Ions must not only flow to facilitate the electron transfer process at the interface (see figure 2(b)), but must also accumulate within the pseudocapacitive region so as to prevent Coulomb explosion and maintain charge neutrality [26,42] where V is the time-evolving electrostatic potential across the system (see figure 2(b)).The electron mobility (µ e ) is obtained from the electron diffusion constant via the Einstein relation where T is the system temperature and k B is Boltzmann's constant.The electric-field dV/dx dependent term in equation ( 6) provides the electron drift contribution, while the electron concentration dρ e /dx dependent term in equation ( 6) provides the electron diffusion contribution.Throughout the charging process ρ e is evolving in time and gives rise to the Ragone behaviour that will be discussed.Similarly, the screening counter-ion flow can be captured in terms of two continuity equations with c + and c − being the positive and negative supporting counter-ion concentrations, respectively.Again, there are both drift and diffusion contributions to the ion flow, which can be seen by comparing equations ( 8) and ( 9) with equation ( 6).For the sake of simplifying the analysis, both ions are assumed to have the same diffusion constant D ion and thus also the same mobility µ ion = qD ion /k B T. Notably, we assume periodic bulk properties transverse to the transport plane (perpendicular to the terminals in figures 1(b) and 2), which allows one to only track drift-diffusion changes along the x-direction in figure 2. Though this does not admit complex 3D structuring, of which a near infinite multitude of possibilities exist [47][48][49][50], it does allow for physically tractable insights that allow one to explore the general characteristics of pseudocapacitive power performance as per the stated goal in figure 1(a).Finally, to solve the continuity equations detailed in section 2.2, one needs to know the electrostatic potential V, which can be obtained from the electric-field (E) via And the electric-field (E) is further obtained from all the system charges ρ where ρ plate accounts for the screening charges on the terminal plates needed to maintain a fixed bias across both terminals in figure 1(b)-e.g. the maximum bias V d discussed in section 2.1.The system permittivity is given by ϵ.Hence, the potential drop and all such spatially dependent equations are solved across a two-terminal system.Following the description in figure 1(b), during charging on one side electrons are added while on the other they are removed [26].

Ragone approach
The first step to computing a Ragone plot for such a system begins with defining the current contributions.
The majority of the current flow is Faradaic and can be tracked as where we integrate only over the charge added on one side across half the terminal separation distance L plate as indicated in figure 1(b).There is also a lesser current contribution due to EDL charging which can be expressed as where L int is some distance into the metallic terminal sufficient to capture all its screening electrons and ρ plate is comprised of the charge stored specifically on the terminal plates (see figure 2(b)).In both equations ( 13) and ( 14), integrating over the entire capacitive region would yield a net charge of zero, since electrons are taken from one side and placed on the other.The net current is thus J = J ps + J EDL , though for a practically relevant pseudocapacitive coating thicknesses J ≈ J ps is a sufficiently accurate assumption.To understand the underlying mechanism responsible for the Ragone plot behaviour in such a pseudocapacitive system, we have found it easier to express the current in a unitless form as The coating thickness of the pseudocapacitive region is defined as L ps as indicated in figure 2(a).Here we have also made use of a diffusion time constant for Faradaic electron insertion into the pseudocapacitive coating of the form which shall be very useful for determining the fundamental physics limiting power performance within a Ragone plot.It is responsible for providing a unitless time variable t = t/τ e .Now since the current is time dependent, so is the net charge stored.It can be also expressed in a unitless form as Finally, the averaged power density can also be expressed in unitless terms at time t = t/τ e via Indeed, the interpretation of power density varies across electrochemical and electrical engineering communities [32].When the energy storage device is placed in an external circuit, the power deliverable by the device varies depending on the loading profile.A maximum power limit can be achieved when the impedance of the energy storage device matches that of the external load [51][52][53][54].However, the design of external circuit and its loading varies across different application, which makes it difficult to benchmark and compare the performance across various energy storage devices.Therefore, in this manuscript, the device performance definition follows the conventions in the electrochemical energy storage community [33,55], where the energy and power densities are measured from the device itself, without considering the coupling with external circuits (i.e. this manuscript focuses on device engineering, and not circuit engineering).This allows the modelling results to be directly comparable to existing Ragone plots in the electrochemical energy storage community.
We have found this averaged power expression in equation ( 19) (at up to a given time t) to provide a meaningful measure of how long it takes to reach a given quantity of charge storage [8,32,56,57].This is useful because it is a measure of power performance independent of the pseudocapacitive coating thickness (per equation ( 17)).Thus, the resulting Ragone trends can be extracted by plotting P versus Q, with Q normalized to a maximum value of 1 and serving as a measure of the total volumetric charge & energy stored.Meaning, when Q = 0 the system is not storing any charge and, conversely, when Q = 1 the system is fully charged.By plotting Ragone trends, and the underlying currents, in these unitless forms it is possible to separate which trends arise from general physical properties (e.g.electronic versus ionic contributions, and drift versus diffusion contributions) from those that vary due to specific system parameters (e.g.L ps , L plate , D e and D ion ).This separation is very important to understanding the underlying performance limitations and general physical properties as we shall discuss shortly.However, for a given system with a specific energy density E d the Ragone trends can be plotted from these trends through with P and E being the total power and energy density for the chosen system of units (e.g.W l −1 and Wh l −1 respectively).Note, that if E d is defined in Wh l −1 then in the computation of P it must be converted into J L −1 in equation (20).Once general trends are extracted via equations ( 18) and ( 19), we shall then move on to system specific considerations in relation to the targets discussed earlier in the context of figure 1(a).

Results
Due to the existence of electronic and ionic drift-diffusion, such a pseudocapacitive system (see figure 1(b)) can be broadly classified as operating in one of three transport regimes determined by the relative magnitude of their ionic and electronic diffusion constants: (I) D ion ≫ D e , (II) D ion ≈ D e , and (III) D ion ≪ D e .Regime I is an ideal situation where nearly instantaneous ion screening is assumed.Hence, in this first regime the power performance capability of the device is limited by electron transport into the redox-active coating of a given thickness.Though Regime I is not realistically feasible, except through intentionally restricting electron diffusion and promoting ion diffusion, it does allow for an understanding of the distinct physical phenomena due to electrons and ions in the Ragone trends we shall explore in Regime II.Regime II is an optimized situation where electron diffusion is matched with ionic drift-diffusion to ensure charge balance [58][59][60][61][62]. Regime III is an unfavourable situation where the ionic drift-diffusion is severely restricted, leading to highly unfavourable resistive performance of the device and shall therefore not be considered in this study.Of course, there are shades of performance between these regimes, but general physical insights can be most easily extracted by considering these limiting cases.In this section, we explore the physical properties of Regimes I and II towards this aim.

Regime I: ideal ionic drift-diffusion
In Regime I (D ion ≫ D e ) the normalized current behaviour ( J) on a log-linear scale demonstrates an initial rapid decay followed by a longer time-scale steady decay (see figure 3(a)).Here we make the assumption that J ≈ Jps , meaning that the double layer contribution to overall charge storage is much smaller than that within the pseudocapacitive coating and does not impact on the Ragone plot behaviour at practically useful charging percentages.This is because the pseudocapacitive coating is sufficiently thick such that its charge storage properties far exceed that of the EDL layer.However, as the time scale in figure 3(a) is normalized by the electron diffusion time constant (i.e.t = t/τ e ) all current plots for various thickness L ps overlap to provide the same trend.Thus, the current decay physics is independent of the coating thickness (L ps ) and can be understood in terms of general electron diffusion properties.Meaning, in Regime I the coating thickness (L ps per τ e in equation ( 17)) will determine the scale of the Ragone plot magnitudes but not its overall trends.Nevertheless, a device with a thicker nanoparticle coating (larger L ps ) will take a longer time to charge, hence its current and charging behaviour will persist over a longer period of time (because it will have a larger electron diffusion constant τ e ).Importantly, electron charge transport properties in Regime I can arrived at by solving equation ( 6) in the approximate form which follows from assuming 'instantaneous' screening from the counter ion densities c + and c − .Meaning the field at the electrode plate (or any subsequent injected pseudocapacitive charge) is screened so rapidly by the supporting electrolyte ions, that the field within the pseudocapacitive region is negligible during the charging process (see figure 2(b)).This may seem spurious, but shall have important relevance for the Regime II (when D e = D ion ) as will be explored below.For the same reason at the iterative timescale over which the continuity equations are solved one can assume that all changes in ρ e are immediately screened by counter-ions.This means that no net internal field will appear between the terminals (per equations ( 10)-( 12)) apart from the sharp interfacial over-potential V int as indicated in figure 2  current tends in figures 3(a) and (b) begin to inflect downwards further due to the limited diffusion space-as can also be seen in figure 3(c).
In both figures 3(a) and (b) an inset (dot-dashed curves) comparison to regular EDL charging is provided, where clear exponential decay differences exist as compared to a pure pseudocapacitive filling current.These trend differences persist in the charge filling trends ( Q) given in figure 3(d).Juxtaposed against the pseudocapacitive filling trends in Regime I (solid figure 3(d)), one can find the classical capacitor filling trend Q = [1 − exp(− t )] with the same time constant t = τ e t (dot-dashed figure 3(d)).Although the two filling trends ( Q) are similar, they are not identical.These distinctions become clearer in the Ragone plot for Regime I (see figure 4), which encapsulates diffusive trends present for a monolithic pseudocapacitive coating: (1) before filling up to the coating boundary wall at L ps (t < τ e , early diffusion in figure 3(c)); and (2) after reaching the coating boundary wall (t > τ e , late diffusion in figure 3(c)).Prior to filling the pseudocapacitive coating up to its boundary (at L ps ) the Ragone behaviour decays linearly on a log-log scale, which corresponds to the point source diffusion of electrons as can be seen in the guiding dashed line that follows this same trend (see figure 4).Then at t = 1, which corresponds to t = τ e when the coating boundary at L ps is approximately reached, there is a downward inflection in the Ragone plot for Regime I (see figure 4 solid lines).Again, regardless of the L ps magnitude all such plots can be normalized to these same trends-this is demonstrated by the many overlapping coloured lines in figure 4 at various L ps values.By comparison the classical capacitor filling trend given by Q = [1 − exp(− t )] results in the dot-dashed line given in figure 4. The two trends only approach each other when filling is largely completed (around Q > 0.8), and differ substantially beforehand due to the point source electron diffusion physics that governs the early stages of pseudocapacitive filling in Regime I (see figure 3(c) and the inset to figure 4).

Regime II: comparable electronic and ionic diffusion
Of course assuming D ion ≫ D e to such a degree that ion diffusion is 'nearly instantaneous' from the perspective of electrons is not practically achievable without intentionally restricting electron diffusion-an undesirable property in terms of power performance.However, it does serve an important role in interpreting the underlying power-performance physics in the optimized regime where D ion = D e .Follow the results of figures 3 and 4, to store more energy the most obvious design approach is to increase both the pseudocapacitive coating thickness L ps monolithically and corresponding plate terminal separation distance L plate simultaneously such that L ps ≈ L plate /2 for large thickness-as shown in figure 5  thicknesses of: 10 nm, 30 nm, 50 nm, 70 nm, 90 nm, 100 nm, 200 nm, 300 nm, and 400 nm, corresponding to the lines with colour red through to blue, respectively.The graph of 10% coating thickness from Regime I result is plotted in purple colour as comparison, and the regular capacitor is plotted as black dot-dashed line in (e) and (f).All results are for a charge density of ρe,max = 100 W l −1 in the coating region separated by a 100 nm thick separating region.
competing with the transfer of electrons per equations ( 2), ( 8) and (9).Once V int is fully formed, via an EDL layer in a practical system, the electron diffusion properties dominate as can be seen through the very similar Q( t ) plots in figures 5(e) and 3(c).This is also evident upon comparing the Ragone plot trends in figures 5(f) and 4, which also show the same limiting behaviour towards electron diffusion.What is even more striking is that electron diffusion becomes more dominant as the pseudocapacitive coating thickness increases towards L ps ≈ L plate /2, meaning the thicker the pseudocapacitive coating the more it behaves like Regime I even when D ion = D e .This can be seen by comparing figures 4 and 5(f).
The origin of this behaviour can be understood in terms of two competing time constants determined by the two length scales L plate and L ps .The time scale for the ionic response is determined fundamentally the resistance to ionic transport and the capacitance which the diffusing ions must fill.The most basic ionic capacitance is the EDL capacitance, which must be fully charged such that V int is saturated (as shown in figures 2(b), 5(b) and (c)), and the transport process is then thereafter dominated by electron injection and diffusion per figure 3.If the total EDL capacitance of the junction is C EDL then the associated charging time constant is where ϱ ion is the ionic resistivity as determined by the ion carrier concentration and diffusion constant (D ion ) discussed earlier in the context of equations ( 8) and ( 9).Now the typical per unit area (A) capacitance of an EDL is fixed around C EDL,A ≈ 40 µF cm −2 [63].Thus, it is the length contribution of L plate , through the ionic resistance R ion = ϱ ion L plate /A, which dominates the scaling behaviour of τ ion as the terminal plate separation distance is increased (see figure 5(a)).Now, if we simultaneously scale both the terminal separation distance and pseudocapacitive coating thickness, then by equations ( 17) and (23) we have the two time constants scaling as τ e ∝ L 2 ps and τ ion ∝ L plate .Moreover, since L ps ≈ L plate /2 in this scenario where L ps is continually thickened monolithically, with thicker coatings the pseudocapacitive τ e ∝ L 2 ps will become more dominant in the system behaviour.This explains why the Ragone trends in figure 5(f) approach closer to those in figure 4 as the pseudocapacitive coating thickness increases (see figure 5(a)).Due to this behaviour it is advantageous to limit the pseudocapacitive coating thickness to such a degree that τ e does not become too excessively large and impractical.However, if we assume a targeted power performance of 10 4 W l −1 in which Q ≈ 0.8 (∼80% charged, which occurs around t ≈ τ e in figure 5(f) then for a target energy density of 100 Wh l −1 a diffusion constant of only D e = D ion ≈ 5 × 10 −11 m 2 s −1 will suffice.These estimates can be obtained by combining equations ( 17) and (20).Based on the analysis in [26], this should be practically achievable and could plausibly provide electron-diffusion-dominated Ragone trends which meet the desired metrics summarized in figure 1(a) [26].
On the other hand, due to non-uniformity in the coating process it may not be possible to achieve sustainable uniform electron diffusion/transport across such large thickness (L ps ≈ 50 µm)-e.g. even sub-micron voids & gaps in the coating will impede electron diffusion [26].This then raises the question of what minimum pseudocapacitive coating thickness one might apply while still obtaining a dominant pseudocapacitive energy storage contribution?The minimum pseudocapacitive coating thickness one can employ is dictated by the necessity to provide an energy storage boost beyond purely EDL chargingotherwise it would serve no purpose beyond a regular supercapacitor (see figure 1(a)).Let us first assume that a given pseudocapacitive coating has a per unit area capacitance of Furthermore, let us again assume a pseudocapacitive electron density storage of ρ e,max = 2.25 × 10 21 cm −3 (or 100 Wh l −1 ) at an applied bias of V d = 5 V. Thus, per equations ( 4) and ( 24) we obtain a L ps,min = C EDL,A /C ps ≈ 2.8 nm when C EDL,A ≈ 40 µF cm −2 [63].If we further employ the metric that the pseudocapacitive charge storage contribution should be substantially greater (e.g.>25×) than that already provided by an EDL interface, then a coating of L ps = 100 nm should serve as a reasonable lower bound.This, of course, is for an assumed pseudocapacitive electron energy density storage of 100 Wh l −1 .Smaller thicknesses might be employed as the energy storage density increases [26].Nevertheless, as outlined in [26], a target storage density of 100 Wh l −1 is likely in the reasonably achievable range of quantized capacitance-though not necessarily at the fundamental limit.Now, since there is minimum and a maximum range concerning desirable pseudocapacitive thickness, let us work within this range to see how such the charging of such a thickness behaves when L ps is held fixed and L plate is scaled (see figure 6(a)).It is necessary to explore these trends to estimate how such coatings might behave if they were to act together in an overall operational pseudocapacitive region approaching or exceeding 50 µm in thickness [30,[34][35][36].For this study let us assume that there is some minimum realistic thickness on the separator region which lies around 0.5 µm.Of course, thicker separators can be explored, but this provides a reasonable juncture for observing the transition from electron-dominated to ion-dominated diffusion as the separation distance between coatings (fixed at L ps = 100 nm) increases as shown in figure 6(a).In figure 6(b) we see that the current trends ( J) show a more lagging time-dependent evolution dominated by ion diffusion as L plate successively increases from 800 nm to 100 µm.The same behaviour can be seen in the charge filling ( Q) plots in figure 6(c), with ion-dominated diffusion taking over once the filling rate falls below the classical capacitor Q = [1 − exp(t/τ e )] trend shown as a dot-dashed line in figure 6(b).This roughly occurs once the ionic diffusion time constant (τ ion ) exceeds the electron diffusion time constant (τ e , which is now fixed since L ps is fixed at 100 nm).The most consequential impact is on the Ragone characteristics in figure 6(d), which transition to a more resistive behaviour as explored in the early studies by Conway [55][56][57].Meaning, the rather sloped Ragone behaviour facilitated by rapid electron injection in figure 4 transitions to a drooping electron-diffusion-dominated resistance characterized by delayed electron injection, as the interfacial potential (V int ) develops, in figure 6(d).Just as in a regular ion-diffusion-dominated EDL capacitor, the time constant for the ionic response scales as τ ion ∝ R ion where R ion = ϱ ion L plate /A as stated above.Thus, in the ion-diffusion-dominated regime τ ion ∝ L plate and this can be seen directly in the charge filling timescale as plotted in figure 6(e).Here we plot the time it takes for the 100 nm thick pseudocapacitive coating to fill to Q = 0.8 (80% charged) as a function of the plate separation (L plate ).While small non-linearities can be seen at shorter separations on a log-log scale, the overall trend is strongly linear (τ ion ∝ L plate ) as the plate separation increases towards ∼100 µm (see the inset to figure 6(e)).
The next logical step is to consider how many such coatings would behave when joined together to form a larger charge storage region on each terminal with a combined thickness ⩾ 50 µm on each side (for a total L plate equalling or exceeding 100 µm, see figure 1(b).This corresponds to piling up many such thin coatings with separation distances ranging from L plate ≈ 1 µm through to L plate ≈ 100 µm as depicted in figure 7(a).Within such a domain of operation, electron injection into a given pseudocapacitive coating will be very rapid, limited only by L ps ≈ 100 nm and D e via equation (17), such that the successively increasing ion diffusion timescales in figure 6(e) become dominant as successive paired coatings are spaced further and further apart.Here we are envisioning in figure 7(a) that electrons could rapidly reach each pseudocapacitive coating, one stacked upon another each ∼100 nm thick, while counter ions much diffuse from opposite terminals (or equipotential conductive electron injection points) in the same manner as figure 6.Thus, we The black dot-dashed line is the Ragone response of a regular capacitor and the gold solid line is the Ragone response of a pseudocapacitive device with successive 100 nm coatings layered up to a total thickness of 50 µm per terminal.(c) Schematic for a similar more feasible horizontal scheme with the coatings stacked horizontally, rather than vertically, in a 'deck of cards' configuration-with each 'card' held and conducting at the applied potential.In both configurations, vertical or horizontal stacking, the mean electron diffusion path from a conducting pathway/surface is are asking in figure 7(a) if one could work to fill a layering of N pseudocapacitive coatings, each L ps ≈ 100 nm thick such that L ps,total = N × L ps , what would be the estimated Ragone power performance characteristics?This is effectively arrived at by computing the overall Q performance, per equations ( 19)- (21), by averaging the trends in figure 6 for N successive increasing L plate separations.The resulting normalized Ragone trends for N = 991 layerings, corresponding to a minimal separation of 1 µm and a maximum separation of 100 µm, can be seen in figure 7(b).Here we see charging properties strongly dominated by ionic resistance, with drooping properties early on in the charging process, in direct contrast to the electron-diffusion-dominated trends in figures 4 and 5(f).This is demonstrated by the manner in which Q (solid line) falls below the classical capacitor charging trend of Q = [1 − exp( t)] shown as a dot-dashed line in figure 7(b)-just as it was provided as a reference in the earlier Ragone plots.Now, it is important to note that the layering mechanism proposed in figure 7(a) is likely not easily achievable, with rapid electron conduction shunts spaced every 100 nm and a manner parallel to the terminal plates.It merely serves as a 'thought experiment' to understand the emerging general trends as ion transport becomes more dominant via its resistive contribution (R ion ) while electron transport is made comparatively rapid by combining many thin coating layers.However, as illustrated in figure 7(c) one can construct a horizontal analogue to the layering approach corresponding to a 'deck of cards configuration' composed of conducting 2D materials separated by 2L ps [64][65][66][67][68][69][70][71], which will have similar electron and ion transport characteristics.That is, ion diffusion will be the fastest near where the oppositely charged 'deck of cards configuration' meet and will be the slowest to respond within the interior coated regions deep inside the 'deck of cards' (near the metallic terminals).Likewise, electron diffusion across a distance of L ps will be similarly rapid between two equipotential 'deck of cards' surfaces spaced 2L ps apart-acting as metallic electron conductors/shunts as shown in figure 7(c).The two schemes can be viewed as the transpose of each other, with near identical electron diffusion and similarly resistive ion diffusion limitations.Indeed, one can  [26].The electron and ion diffusion constants are held equal optimized in a range of 10 −15 m 2 s −1 to 5 × 10 −11 m 2 s −1 in the bronze and dashed line bounded regions, respectively.The bronze region diffusion constant estimate is for many layered coatings, each 100 nm thick, subject to an equipotential conducting electron source; while the dashed line bounding region is for a single electron source at the terminal and a thick monolithic pseudocapacitive coating.Comparative regions for other technologies are also shown.
argue that any scheme which includes conducting filaments or sheets such that the mean electron diffusion path becomes much less that the ion diffusion path, meaning L ps ≪ L plate via electron transport assisting conducting filaments/pathways, essentially becomes dominated by ion resistance in the same general manner as provided in figures 6 and 7.The major question, of course, is how any such proposed schemes will perform in practice concerning the desired Ragone performance in figure 1(a).
To this end, let us take the ion-transport-dominated results in figure 7(b) as a best case scenario benchmark for pseudocapacitive coating thickness on the order of 50 µm (i.e.L plate ≈ 100 µm & L ps ≈ L plate /2).By employing the relations in equations ( 19)- (21) we can convert the trends in figure 7(b) to a unit based electron storage density of 100 Wh l −1 and obtain the time constant (τ e ) required to deliver a corresponding power performance of 10 4 W l −1 -note all the results in figure 7(b) are normalized by τ e .This works out to approximately τ e ≈ 6 s, which when employing the relation in equation (17) for L ps = 100 nm provides D e = D ion ≈ 10 −15 m 2 s.Thus, the effect of a successive coating based approach (combined with equipotential conducting electron pathways that maintain L ps ≪ L plate ) is to lower the necessary diffusion constant required to meet the target power density (i.e. 10 4 W l −1 ).This arises because the charging time is now dominated by τ ion ∝ L plate rather than electron diffusion and τ e ∝ L 2 plate /4 (when a single monolithic pseudocapacitive coating is applied as discussed earlier in the context of figures 4 and 5).Recall that the earlier electron-dominated diffusion trends in figure 5 Of course, one can argue that the layered approximations resulting from figures 7(a) and (b) may over estimate the ease of ion transport and that other geometries will have different ion flux properties [72][73][74].Moreover, ion-correlations and depletions may play a further role in ion transport [75][76][77][78][79].All of this is certainly important, but digresses from the general physics that is concerned with estimating whether a realistically achievable diffusion constant can provide a targeted performance of 10 4 W L −1 at 80% charging for a overall pseudocapacitive coating thickness of ∼50 µm.What we have in hand are two extreme estimates regarding the diffusion constant needed achieve to provide this power performance metric.The upper bound estimate suggests that D e = D ion ≈ 5 × 10 −11 m 2 s −1 is needed to achieve this metric, while the lower bound estimate suggests that D e = D ion ≈ 10 −15 m 2 s −1 can achieve this metric.If we are conservative in our estimate, between these extremes, we can suggest D e = D ion ≈ 10 −12 m 2 s −1 as an achievable goal to reach this power performance.That is, even if geometric and ion-correlation interactions further complicate ion diffusion, they should not raise the necessary diffusion constant above this range-otherwise geometric structuring and electron conducting pathways, for example, would serve no beneficial ion transport purpose [80][81][82][83][84][85].For electron diffusion D e ≈ 10 −12 m 2 s −1 is certainly achievable as discussed in [26].Likewise, for ion diffusion performance in the range of D ion ≈ 10 −12 m 2 s −1 is also reasonably achievable [62,[86][87][88][89][90][91][92][93][94].The long and the short of this analysis is that the pseudocapacitive batteries employing quantized capacitance should, theoretically, be able to achieve the targeted power performance set out in figure 1(a) with diffusion constants that are plausible and practical coating thickness that are around 50 µm on each terminal as shown in figure 1(b) [95][96][97][98][99].This is because the diffusion constants, as laid out by this analysis, needed to achieve this performance are feasible.Though some sort of layering or conducting filamentary construction whereby electron diffusion does not become so limiting, such that the mean electron diffusion path is relatively small (L ps ≪ L plate ), would only prove beneficial to performance by relying more ion transport that scales better with system size (τ ion ∝ L plate ).Moreover, the nature of the dominating diffusion species (electrons or ions) can be determined from the Ragone plot characteristics.When resistive ion transport dominates the Ragone response should resemble the drooping bronze curve in figure 8, while when electron diffusion dominates the system response (e.g. when L ps ≈ L plate /2) one should obtain a response more similar to the slopped dashed curve in figure 8.

Conclusion
In this study we theoretically explored the power performance capabilities of a pseudocapacitive battery constructed based on the principle of quantized capacitance.Taking as our metric 80% charging of a 100 Wh l −1 storage density at a rate of 10 4 W l −1 , we were able to estimate the operable range of diffusion constants that could attain this metric performance when positive and negative terminal coatings were each set at a practically relevant ∼50 µm thickness (for a total terminal separation distance on the order of 100 µm).The ultimate metric of performance was taken to be the resulting Ragone plot of a given configuration.First, we explored the properties in the charging regime where the pseudocapacitive coating was monolithic, such that electron injection only occurred at the terminal ends and the pseudocapacitive coating thickness was approximately half the terminal separation distance (L ps ≈ L plate /2).The performance in such a system was shown to depend largely upon the electron diffusion constant (D e ) regardless of whether one operated in Regime I where the ion diffusion constant was much greater than electron diffusion constant or Regime II where the ion diffusion constant equals the electron diffusion constant (D ion ≫ D e or D ion = D e ).For such a monolithic coating approach, the Ragone behaviour was shown to provide a gradually sloping downwards trend dominated by electron diffusion that terminated once the coating was largely filled (e.g.⩾80 %).Moreover, for monolithic coating with a thickness of 50 µm, equivalent ion and electron diffusion constants of ∼5 × 10 −11 m 2 s −1 were shown to be sufficient to achieve the aforementioned power metric.Next, we explored the charging characteristics of successive thin film pseudocapacitive coatings on the order of 100 nm thick that could each be provided with a source/shunt of electrons held at the applied potential.This resulted in a drastically reduced electron diffusion time constant, such that the charging characteristics became dominated by ionic resistance.The Ragone behaviour in this system was shown to provide a drooping behaviour at low energy densities, followed by a rise towards the targeted power performance metric with a much smaller diffusion constant value of D e = D ion ≈ 10 −15 m 2 s −1 (for a layering of coatings approximately 50 µm thick).Thus, system power performance generally improves when the mean path of electron diffusion is reduced-e.g.through the introduction of conductive materials such as in a 'deck of cards configuration' [66,[100][101][102].These differing diffusion constant estimates arise because the ionic transport time constant scales approximately as τ ion ∝ L plate for equipotential conducting surfaces, in a similar manner to that of a regular parallel plate or EDL capacitor, while the electron diffusion time constant scales as τ e ∝ L 2 ps .Taking this range of diffusion estimates into account our analysis suggests that a conservatively targeted diffusion constant of D e = D ion ≈ 10 −12 m 2 s −1 between these extremes should be able to meet the targeted performance metrics.Meaning, a pseudocapacitive battery constructed to typical dimensionalities (L plate ≈ 100 µm with a redox-active coating thickness approaching 50 µm on each terminal) should be able to achieve good energy density (∼100 Wh l −1 ) and strong power performance (∼ 10 4 W l −1 ) with diffusion constants in the range of D e = D ion ≈ 10 −12 m 2 s −1 , the construction of which should include electrically conducting pathways/filaments such that the average electron diffusion distance is comparatively small compared to the ion diffusion distance.However, the geometry and composite selection of filamentary materials (e.g.conductive 2D materials) which best promotes ion diffusion and limits the electron diffusion time constant is left for a detailed future study.Future work should consider the study of different active materials requiring much more complex and material-specific modelling to include effects such as pore size, pore accessibility, interphase structure, and diffusion pathways.The major takeaway from this theoretical study is that the 'holy grail' of combined good energy and power density performance is plausibly achievable through an energy storage system employing quantized capacitance-i.e.pseudocapacitive batteries.Future work should focus on the experimental validation of these estimates.

Figure 1 .
Figure 1.(a) (Adapted from [26]) Ragone plot depicting the estimated position of quantized capacitance and its Ragone curvature behaviour, which is explored in this paper regarding power performance (indicated by a question mark).The arrow indicates the goal of achieving high energy and power densities.(b) Schematic depicting the structure of an energy storage device employing quantized capacitance, with the active material coating thickness indicated in grey and the separation between charging terminal plates indicated as L plate .Reprinted figure with permission from [26], Copyright (2022) by the American Physical Society.

Figure 2 .
Figure 2. (a) Electron diffusion (De) into the conducting nanoparticles for energy storage that relies on interfacial electron transfer rate (ket) and inter-particle tunnelling rate (k ip ).(b) Drift-diffusion of the ionic species (D ion ) in the electrolyte to screen off the interfacial potential (V).

Figure 3 .
Figure 3. Normalized current versus time ( J vs. t) plot on (a) log-linear and (b) log-log scales that demonstrates a non-linear relationship for active material coating thicknesses ranging from 10% to 50% of the device length.Note that panels (a) and (b) only show one plot because all the plots overlap with each other due to normalization with the electron time constant (τ e).Point-source Gaussian diffusion (black dashed line) is included for comparison.(c) Electron storage in the nanoparticle coating over time.(d) Total charge storage as a function of time compared to the point-source Gaussian diffusion (black dashed line) and a regular capacitor.
(b).The secondary consequence is that one can approximate the injection of electronic charge as a point-source Gaussian diffusion process governed by injection at x ≈ 0 into the pseudocapacitive region in the limit L ps → ∞.Its resulting current trends (L ps → ∞) are provided as a dashed line in the figures 3(a) and (b), where one can see that the coating filling trend (solid line) rolls off from the point source diffusion trend (dashed line) at around t = τ e .This transition point corresponds to the diffusing electron density front reaching the finite physical boundary of x = L ps as shown in figure3(c).At this point (t = τ e or t = 1), the

Figure 4 .
Figure 4. Normalized Ragone plot of pseudocapacitive batteries operating in Regime I in comparison with point-source Gaussian diffusion (dashed black line).Note that the plots with nanoparticle coating thicknesses ranging from 10% to 50% of the device length overlap each other due to normalization with the electron diffusion time constant (τ e) and coating thickness (Lps).The performance of a capacitor with a time constant of τ e is shown for comparison as a dot-dashed line.
(a).The corresponding potential drop (V, blue) and stored pseudocapacitive charges (ρ e , red) are shown in figures 5(b) and (c) for two very different coating thicknesses when D ion = D e .Now, if we look at the pseudocapacitive charging current Jps in figure 5(d) (again we are now considering Regime II where D ion = D e ), it looks very similar to the trend for Regime I in figure 3(b) (the earlier case where D ion ≫ D e was considered).The only difference is a slight tail at very early times, which is due to the initial build up of V int in figure 2(b) via ion diffusion

Figure 5 .
Figure 5. (a) Schematic showing a gradual increase in nanoparticle coating thickness (Lps) applied monolithically.The potential bias applied to the device (blue line) and the corresponding charge distribution (red line) for (b) a thin coating of 10 nm and (c) a thick coating of 400 nm.(d) Normalized current density, (e) total charge storage as a function of time, and (f) normalized Ragone plot as a function of increasing Lps.Plots (d) to (f) are compared to the point-source Gaussian diffusion (black dashed line) at Lpsthicknesses of: 10 nm, 30 nm, 50 nm, 70 nm, 90 nm, 100 nm, 200 nm, 300 nm, and 400 nm, corresponding to the lines with colour red through to blue, respectively.The graph of 10% coating thickness from Regime I result is plotted in purple colour as comparison, and the regular capacitor is plotted as black dot-dashed line in (e) and (f).All results are for a charge density of ρe,max = 100 W l −1 in the coating region separated by a 100 nm thick separating region.

Figure 6 .
Figure 6.(a) Schematic showing a gradual increase in plate separation while keeping the pseudocapacitive coating thickness constant.(b) Normalized current density of increasing plate separation.(c) Normalized charge density accumulation.(d) Ragone plot trends for the same plate separations juxtaposed against the Gaussian diffusion model (black dashed line).(e) Timescale for reaching 80% charging (t 80% ) as the coating separation increases, with the inset showing linear scaling at large plate separations.The plate separation ranges from 0.8, 1, 2, 4, 6, 8, 10, 20, 40, 60, 80, to 100 µm for a 100 nm fixed coating thickness, corresponding to the dark red to orange lines indicated by the arrows in (b) to (d).The 10% coating thickness result from Regime I is plotted in purple colour as a comparison, and the regular capacitor trend is plotted as a black dot-dashed line.

Figure 7 .
Figure 7. (a) Many layered pseudocapacitive coatings, where electron injection is easily facilitated by a conducting pathway/shunt to the start of each coating held at the applied potential.(b) Resulting Ragone plot for many such 100 nm coatings filling together, normalized by the coating time constant τ e.The black dot-dashed line is the Ragone response of a regular capacitor and the gold solid line is the Ragone response of a pseudocapacitive device with successive 100 nm coatings layered up to a total thickness of 50 µm per terminal.(c) Schematic for a similar more feasible horizontal scheme with the coatings stacked horizontally, rather than vertically, in a 'deck of cards' configuration-with each 'card' held and conducting at the applied potential.In both configurations, vertical or horizontal stacking, the mean electron diffusion path from a conducting pathway/surface is

Figure 8 .
Figure 8. Ragone plot showing the theoretical maximum power density performance of a device with a typical coating thickness of ∼50 µm at each terminal and a theoretical maximum energy density of approximately 100 Wh l −1 as reported in[26].The electron and ion diffusion constants are held equal optimized in a range of 10 −15 m 2 s −1 to 5 × 10 −11 m 2 s −1 in the bronze and dashed line bounded regions, respectively.The bronze region diffusion constant estimate is for many layered coatings, each 100 nm thick, subject to an equipotential conducting electron source; while the dashed line bounding region is for a single electron source at the terminal and a thick monolithic pseudocapacitive coating.Comparative regions for other technologies are also shown.
(f) provide a power performance of 10 4 W l −1 when D e = D ion ≈ 5 × 10 −11 m 2 s −1 & L ps ≈ 50 µm.Both trends are plotted in figure 8 for these assumed diffusion properties towards a targeted power performance of 10 4 W l −1 .