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Towards a Pseudocapacitive Battery: Benchmarking the Capabilities of Quantized Capacitance for Energy Storage


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Despite being capable of very fast charging, the pseudocapacitive properties of electrochemical capacitors still require significant research to attain energy densities comparable to that of batteries. Herein we discuss and theoretically benchmark the physics of quantized capacitance as a Faradaic charge storage mechanism, providing near “ideal” pseudocapacitive properties in the context of batterylike energy storage. Through careful electrolyte and reactant engineering, our physical analysis suggests that this less explored “pseudocapacitive battery” mechanism could provide power densities of approximately 10 4 W/L combined with volumetric energy densities in the range of 100 Wh/L (or potentially greater). These benchmarks are arrived at though a comprehensive analysis of two-dimensional (2D) graphitic nanoparticles considering the impact of solvation, electron-electron interactions, and electron transfer processes. In general, our findings indicate that 2D nanomaterials exhibiting quantized capacitance provide a promising and underexplored physical axis within electrochemical capacitors towards realizing very fast charging at energy densities comparable to that of batteries.
(a) Comparative electronic structure plots against an electrolyte stability window (green) situated between −2 and −7 eV below vacuum. The left panel shows the electronic structure of C 60 , the middle panel that of graphene, and the right panel that of BrC 8 decorated graphene. A dashed line provides the alignment of all electronic structure plots to the Fermi energy (E f ) of graphene. The Fermi energy of single-layer BrC 8 lies below that of graphene, due to the loss of electrons to Br atoms, but above the stability limit of the electrolyte window. (b) A schematic depicting the position of the open circuit potential (V OC ) within the electrolyte stability window (shaded in light blue) dictates the electrode potentials (|V anode | and |V cathode |) and the maximum cell potential V max for a system with symmetric electrodes. This schematic assumes the simple case that V OC lies at the center of the electrolyte stability window with the equally distributed current peaks (shaded in gray) for electron removal at the positive terminal and electron addition at the negative terminal for different charge states on the graphitic nanodisks during a charging process. Here, the equal peak distribution (U + = U − ) on each terminal is achieved by nanodisks of similar radius on both sides. The inset shows the definitions of anode and cathode for a supercapacitor cell under the charging process [97]. (c) A schematic for a more frequently observed V OC that lies closer to the limits for the positive electrode. To avoid wasting the wider potential window on the negative electrode, the nanodisks can be engineered to have a smaller radius for a larger U − as compared to U + .
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PRX ENERGY 1, 013007 (2022)
Featured in Physics
Towards a Pseudocapacitive Battery: Benchmarking the Capabilities of
Quantized Capacitance for Energy Storage
Yee Wei Foong,1,*Javad Shirani,1Shuaishuai Yuan,1Christopher A. Howard,2and
Kirk H. Bevan 1,3,
1Division of Materials Engineering, Faculty of Engineering, McGill University, Montréal, Québec H3A 0C5,
2Department of Physics and Astronomy, University College London, London, United Kingdom
3Centre for the Physics of Materials, Department of Physics, McGill University, Montréal, Québec H3A 2T8,
(Received 14 January 2022; revised 22 April 2022; accepted 11 May 2022; published 23 June 2022)
Despite being capable of very fast charging, the pseudocapacitive properties of electrochemical capac-
itors still require significant research to attain energy densities comparable to that of batteries. Herein
we discuss and theoretically benchmark the physics of quantized capacitance as a Faradaic charge stor-
age mechanism, providing near “ideal” pseudocapacitive properties in the context of batterylike energy
storage. Through careful electrolyte and reactant engineering, our physical analysis suggests that this
less explored “pseudocapacitive battery” mechanism could provide power densities of approximately
104W/L combined with volumetric energy densities in the range of 100 Wh/L (or potentially greater).
These benchmarks are arrived at though a comprehensive analysis of two-dimensional (2D) graphitic
nanoparticles considering the impact of solvation, electron-electron interactions, and electron transfer pro-
cesses. In general, our findings indicate that 2D nanomaterials exhibiting quantized capacitance provide a
promising and underexplored physical axis within electrochemical capacitors towards realizing very fast
charging at energy densities comparable to that of batteries.
DOI: 10.1103/PRXEnergy.1.013007
Electrochemical capacitors combine both electric dou-
ble layer (EDL) and Faradaic mechanisms to maintain
high power densities, but at energy densities that exceed
the performance of purely EDL-based capacitors. This
improvement is often accomplished by utilizing redox-
active nanoparticles at the electrode surface, which store
additional electrons in a Faradaic manner that mimics EDL
charge storage “pseudocapacitively.” It is the ultimate goal
of this effort to maintain the high power performance
of electrochemical capacitors while pushing towards the
energy density regime typically occupied by batteries [1,
2]. A great deal of research is currently focused on realiz-
ing a high performance pseudocapacitance energy storage
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license. Fur-
ther distribution of this work must maintain attribution to the
author(s) and the published article’s title, journal citation, and
enabled by nanomaterials [36]. However, despite exten-
sive experimental activity, the physics which underlie a
pseudocapacitive response in a given nanomaterial system
are not well understood [79]. Recently, it was proposed
that suitably engineered conducting nanoparticles might be
tailored to exhibit a near ideal pseudocapacitive response
through the use of “quantized capacitance”—an observ-
able Faradaic mechanism in nanoparticles arising from
electron-electron interactions related to Coulomb blockade
[1017]. However, the energy and power density capa-
bilities of “quantized capacitance” have yet to be fully
explored. Herein, we seek to theoretically benchmark the
maximum energy and power density storage capabilities
for an electrochemical system making use of quantized
capacitance and its promising pseudocapacitive features.
Through our analysis we are able to show that quantized
capacitance can, theoretically, provide the combination of
high power density and energy density long sought by the
field [1].
In Fig. 1the approximate power and energy density
performance of various energy storage technologies are
provided in the form of a Ragone plot. The highest
power density is provided by conventional capacitors (red,
Fig. 1), though they suffer from very low-energy storage
2768-5608/22/1(1)/013007(17) 013007-1 Published by the American Physical Society
YEE WEI FOONG et al. PRX ENERGY 1, 013007 (2022)
Power density (WL–1)
(Wh L–1
10−2 10010 210 4
Fuel cells
FIG. 1. (Adapted from Ref. [1].) A Ragone plot depicting the
volumetric power and energy densities of various energy storage
devices. Theoretical energy storage benchmarks for a “pseudo-
capacitive battery” operating via quantized capacitance (gold
region) are discussed and presented in this manuscript.
density. In supercapacitors, the EDL mechanism is tailored
through multiscale nanostructuring to maintain a compar-
atively high power density while extending towards vol-
umetric energy densities of the order of 10 Wh/L (green,
Fig. 1)[18,19]. On the other hand, conventional batteries
provide much better energy storage than supercapacitors
but typically at a much lower power density [2022]. The
driving impetus, behind engineering a pseudocapacitive
component within an electrochemical capacitor, is to main-
tain the fast charging properties of supercapacitors while
extending performance towards the energy densities cur-
rently occupied by batteries [36,18,19]. This long stand-
ing nexus is shown in gold in Fig. 1, our specific aim is to
assess the degree to which “quantized capacitance” might
be engineered to yield both high power and energy den-
sities within this region resulting in a “pseudocapacitive
Our investigation is motivated by recent promising
developments in the usage of graphitic nanoparticles,
where they have been utilized electrochemically to great
effect by tuning both their dimensionality and laminate
packing [23,24]. It has also been shown that carbon nanos-
tructures can store high densities of electrons [2328].
Additionally, from a pseudocapacitive perspective graphite
or graphene is intriguing, since it is a bulk conduc-
tor, which through sufficient nanostructuring can provide
“quantized capacitance” charging states that are acces-
sible electrochemically [12,2931]. Moreover, graphitic
nanoparticles can be resolved down to one atomic layer
and such that all atoms equally participate in charge
storage. That is, there is no internal region in such a
two-dimensional (2D) material and therefore the charge
storage as a function solely of the nanoparticle volume is
maximized—compared to say a conducting sphere, where
net charge aggregates towards the surface. Driven by
these developments we utilize the proposed “quantized
capacitance” energy storage scheme in Fig. 2(a) to bench-
mark its performance within the gold region highlighted
in Fig. 1. Though quantized capacitance has been inves-
tigated experimentally at dilute concentrations, exhibit-
ing intriguing multiple redox and amphoteric properties
[12,29,31,32], its ultimate theoretical potential as an
energy storage mechanism at high packing densities has
not been explored.
The fully charged configuration within this storage
scheme is provided in Fig. 2(a), where the negative elec-
trode holds electrons (e) taken from the positive electrode
to leave behind holes (h+, the absence of an electron).
Screening counterions must be present within both elec-
trodes in equal concentration to their stored charges, as
shown in blue and red in Fig. 2(a), to prevent the onset
of Coulomb explosion [33,34]. Likewise, the discharg-
ing state is illustrated in Fig. 2(a), where electrons placed
on the negative terminal move back over to the positive
terminal and counterions diffuse accordingly in the oppo-
site direction. Apart from the use of quantized graphitic
nanoparticles, the scheme in Fig. 2(a) is operationally quite
similar to that of a redox-polymer battery as juxtaposed
in Fig. 2(b) [22,3537]. The two schemes are analogous
in several respects: (1) redox processes occur throughout
both the positive and negative terminal charging media;
(2) electrons are stored on the negative terminal and holes
on the positive terminal; (3) the diffusion of electron and
holes is facilitated by intersite electron transfer; and (4)
counterions are allowed to freely diffuse to prevent the
onset of Coulomb explosion [22,33,3537]. However, the
use of quantized nanoparticles enables two key additional
features: (1) the charge stored at a given voltage can be
tuned through dimensionality engineering; and (2) multi-
ple redox events occur at each site through the use of quan-
tized capacitance [10,29,31]. This latter difference is key as
it enables a near “ideal” pseudocapacitive behavior in such
nanoparticles, in direct contrast to the peaked voltammet-
ric behavior found in a redox-polymer battery—contrast
the lower green voltammogram in Fig. 2(a) with the lower
orange voltammogram in Fig. 2(b). Indeed, one might con-
sider the proposed mechanism in Fig. 2(a) to be that of a
“pseudocapacitive battery.”
To explore the physical feasibility of such a “quantized
capacitance” storage mechanism, of the form presented
in Fig. 2(a), our analysis is divided into several parts.
First, we explore the volumetric energy density limits
that would be provided by this scheme in Sec. II. This
encompasses a discussion on the general electron stor-
age capabilities in Sec. II A, followed by an overview on
the quantized capacitance redox mechanism in Sec. II B
and its operational voltage tuning capabilities in Sec. II C.
Our volumetric energy density assessment concludes with
a consideration of how electrolyte stability impacts upon
energy storage via this mechanism in Sec. II D.Thenin
Sec. III we address how the power performance targeted in
Quantized Capacitance
Faradaic and Pseudocapacitive
Redox Polymer Batteries
Faradaic but not Pseudocapacitive
(a) (b)
Charged State
Supporting Electrolyte Carrier Species
Charged State
Supporting Electrolyte Carrier Species
FIG. 2. (a) Operational schematic for a quantized capacitance device similar to (b) (adapted from Ref. [22]) a redox-polymer battery.
Both schemes are depicted in the fully charged state and during discharging. Their corresponding cyclic voltammograms are illustrated
in the bottom panel of each subfigure. In a quantized capacitance device, graphitic nanoparticles with negative charge states are utilized
to store electrons during charging, while graphitic nanoparticles with positive charge states are utilized for electron (e) stripping for
charge counter balancing. Here, the stripping or removal of electrons (to realize positively charged states) is denoted by “holes”
(h+)—just as it is in a redox-polymer battery. This design is provided to explore the possible capabilities of the mechanism; more
advanced multiscale designs to aid charge carrier diffusion and transport will likely be important in any practical implementation
[3840]. Redox-based flow battery analogues may also be similarly applied [4143].
Fig. 1might be achieved. Lastly, in Sec. IV we discuss the
relevance of “quantized capacitance” to similar technolo-
gies (such as redox-polymer batteries) and summarize our
findings. It is important to emphasize that our goal
throughout is to assess the general physical possibility
of such a storage mechanism; not to delve into highly
system-specific electrolyte, packing, or other engineering
YEE WEI FOONG et al. PRX ENERGY 1, 013007 (2022)
A. General energy storage design concepts
Fundamentally, the volumetric energy storage density
of a system is a product of the density at which elec-
trons are stored and the voltage Vat which they are
placed. In a capacitive system this is summarized by
2QV, where Cis the capacitance and Qis
the charge stored (Q=CV)[44,45]. Since we are con-
sidering nanoscale graphene disks in our model system
[see Fig. 2(a)], this energy storage density is then propor-
tional to the number of electrons stored in such nanodisks.
Although our manuscript discusses energy storage design
within the context of a graphitic nanodisk-based system,
the approach discussed below can also be applied to a
range of similar nanomaterials [4649]. However, nan-
odisks are likely advantageous as they utilize a minimal
amount of pseudocapacitive volume to store charge (hav-
ing no “interior region,” for example compared to spherical
Recent measurements have shown that nanoscale
graphitic systems can store one electron for approximately
every ten carbon atoms [23,24]. This achievable ratio,
when applied to graphene or nanodisks thereof, results in
a surface electron storage density of σe4q/nm2—where
q=1.6 ×1019 C is the elementary charge. Although this
electron density is less than the theoretical maximum of
fully intercalated graphite in batteries [5052], it is still
a significant storage density for supercapacitor systems.
To induce quantized capacitance, a nanoparticle must be
separated from other similar particles by a supporting
medium [10,11,17,23,53], since this promotes electron-
electron interactions and enables one to tune the storage
voltage, as will be discussed in Sec. II B. The dielectric
properties of this insulating medium, which is quite often
an electrolyte, also impact upon the voltage storage prop-
erties associated with quantized capacitance [10,17,54]—a
topic to be discussed in depth shortly as well. Thus, when
combining our model graphene nanodisks with a support-
ing medium separating them, which can be as thin as
1nm[10,17,55], one obtains the electron density trend
presented in Fig. 3(a). It is critical that one assess the elec-
trolyte fraction present in the porous electrode because, to
arrive at a plausible energy storage technology, the pack-
ing density of nanoparticles must be increased compared to
existing scientific studies of quantized capacitance [11,29].
This is needed to achieve high volumetric energy densities
via this mechanism, similar to how the molar concen-
tration of redox species should be increased to achieve
higher volumetric energy density in a flow battery. Here we
have assumed an effective thickness for a graphene nan-
odisk corresponding to d0.4 nm, roughly equal to the
spacing between graphite sheets [23,24], and have varied
the total volume from 2 times to 20 times the reactant
(nanodisks) volume. Accordingly, at a storage voltage of
Vd=5 V, one then obtains the volumetric energy den-
sity trends presented in Fig. 3(a) that can be described
Here Lis the thickness of the supporting medium region
relative to the disk region—a parameter obtainable by
summing all the nanodisks as a “single surface” and plac-
ing it atop the volume of the entire supporting medium
normalized to the same surface area [see Fig. 3(b)]. Note
that a factor of 1
4is appended to the energy density
expression in Eq. (1).Thefirst1
2multiplier in this con-
tribution arises from the equal volume of opposite charge
that must be stored at the cathode [see Fig. 2(a)]. The
second 1
2multiplier arises from the manner in which
charge is stored via quantized capacitance, being added
in equal degrees at higher and lower voltages for a given
terminal just like a regular capacitor [10,17,54]—further
discussion can be found in the Supplemental Material
[56]. From Fig. 3(a), it can be seen that when half of
the volume is electroactive, we obtain the upper Ragone
energy density limit of about 250 Wh/L for quantized
capacitance, as shown in Fig. 1. On the other hand,
the energy density is significantly degraded when the
overall volume is 20 times greater than the electroac-
tive contribution—leading to the lower limit provided
in Fig. 1. The higher extreme of about 250 Wh/L is
likely unrealistic and the lower limit is likely impracti-
cal, but arguably intermediate densities around 100 Wh/L
are theoretically achievable, as we will explore further
B. Mechanisms giving rise to quantized capacitance
Our primary energy density assumption in Figs. 1
through 3is that the redox potentials of nanoparticles
exhibiting quantized capacitance can be pushed towards
encompassing a bias window of near 5 V. This is arguably
the maximum achievable bias window for most state-of-
the-art electrolyte systems [5760]. Here we detail how
energy storage at this voltage “limit” might be accom-
plished via quantized capacitance. A detailed description
of quantized capacitance as a pseudocapacitive mechanism
has previously been published in Ref. [10]. Readers who
are familiar with this topic can proceed to Sec. II C and
thereafter, where the main contributions of this manuscript
are discussed. Nevertheless, we briefly describe the quan-
tized capacitance mechanism in this section for the benefit
of a broad readership.
When an electrode is biased towards electron storage in
the manner shown on the left-hand side of Fig. 2(a), the
potential difference will raise the Fermi energy level in
the electrode relative to nanoparticles in the electrolyte,
0.0 2468101214161820
thickness d
Supporting medium
thickness L
Nanoparticle Layers
Total Volume to Reactant Volume
FIG. 3. (a) Electron density as a function of the ratio of total volume to reactant (nanodisks) volume, assuming a surface electron
storage density of 4 q/nm2and the corresponding volumetric energy density at 5 V storage voltage. (b) A schematic that shows the
total volume to reactant (nanodisks) volume can be calculated by taking the ratio of the nanodisk thickness dand the supporting
medium thickness L.
as shown in Fig. 4. This bias will then initiate electron
transfer into the unoccupied electronic states present in
the nanoparticles [see Fig. 4(b)]. Because of the limited
size of the nanoparticles, each electron being added will
experience measurable electron-electron repulsion, lead-
ing to the initial charging energy cost Uothat constitutes
quantized capacitive behavior [1017]. For a nanodisk, this
charging energy cost Uocan be approximately expressed
as [10,61]
Here, ois the permittivity of vacuum, op is the optical
dielectric constant, and ris the nanoparticle radius. The
function F(r)accounts for the average electrostatic poten-
tial across a uniformly charged disk—a detailed explana-
tion of this expression can be found in the Supplemental
Material [56]; see also Ref [61]. After solvent reorgani-
zation the placement of the electron reduces to U, which
includes orientational dielectric contributions present in
liquid electrolyte:
Here ris the relative permittivity of the electrolyte and
λis the heterogeneous reorganization energy [17,62,63].
As a result of these interactions, a voltammetric scan
will exhibit multiple overlapping electron transfer current
peaks separated by U—shown as dashed lines in Fig. 4(a)
[10,16,17,64]. By carefully engineering Uone can physi-
cally tailor the individual redox peaks to sufficient overlap
with each other such that a near-rectangular voltamme-
try profile for pseudocapacitive energy storage behavior is
enabled [solid line in Fig. 4(a)][10].
Importantly, we are considering a tunneling electron
transfer process between the electrode and nanoparti-
cle dispersion, as shown in Fig. 2(a) [17,22]. Hence,
the multiple redox peaks presented in Fig. 4(b) can
YEE WEI FOONG et al. PRX ENERGY 1, 013007 (2022)
Spatial Position
(a) U
FIG. 4. (Adapted from Ref. [10].) (a) Cyclic voltammetry of
quantized capacitance up to a maximum applied potential V,max
on a single electrode that resembles an ideal pseudocapacitive
behavior due to the overlapping electron transfer peaks. (b)
A schematic of a single negative () electrode depicting the
operation of quantized capacitance Faradaic storage via electron
tunneling to each reactant state. Here, the schematic describes
the electron storage at a cathode during the charging process (as
the electrode electrochemical potential μeq is raised). At V,max
the desired electron density is stored. A similar schematic can be
used to understand electron removal at the positive (+)terminal
during the charging process.
each be described within Gerischer-Hopfield theory
[10,63,6567]. Each electron transfer event into a nanopar-
ticle with Nelectrons is described by the oxidation distri-
Dox,N(ε) =1
4πλkBTexp εox,N)2
where εis the single-particle energy found in the
Gerischer-Hopfield framework [10,63,6567]. Likewise,
an electron removal event from a nanoparticle with N
electrons occurs via
Dred,N(ε) =1
4πλkBTexp εred,N)2
where kBis the Boltzmann constant and Tis the temper-
ature. Moreover, εox,Nis the single-particle energy level
of the Nth oxidized state; similarly, εred,Nis the single-
particle energy level of the Nth reduced state. These
single-particle redox levels are then related by [10,53,62]
Here it is assumed that wavefunction quantization contri-
butions to the total energy arising from an electron addition
or removal event are negligible [10,17]. Crucially, we can
engineer λand Uto tune the redox peak placement in a
quantized capacitance system to encompass a target V,max
placement voltage for a given number of electrons [see
Fig. 4(b)]. However, overall storage voltage Vdof the
system envisioned in Fig. 2(a) is determined by the sum
of the maximum potential drop across two such termi-
nals: one biased, as depicted in Fig. 4(b) [forming the
negative terminal in Fig. 2(a)], and the other oppositely
biased for electron removal [forming the positive terminal
in Fig. 2(a)]. The nature of this biasing is a subtle point for
consideration that will be discussed in Sec. II D.
C. Tuning the energy storage voltage
From Secs. II A and II B we arrive at two key met-
rics. First, we would like to arrive at a nanodisk electron
storage density of around 4 q/nm2for the proposed mech-
anism in Figs. 1and 2(a) [23,24]. Second, we need to tune
the charging energy parameter Usuch that this density of
electrons is stored and removed at a bias of about 2.5 V
on a given terminal relative to the fully discharged state
(for a total of about 5 V across both terminals). From Eqs.
(2) and (3) we can see that the solvent dielectric constant
(r) and nanodisk radius (r) are two key physical means
for accomplishing this. In Fig. 5(a) we plot the operating
voltage at a given terminal as a function of the solvent
dielectric constant for several nanodisk radii, all storing
electrons at a density of σe=4q/nm2. The total num-
ber of electrons stored in a given disk is πr2σe; this can
be coupled with Eqs. (2) and (3) to provide the trends in
Fig. 5(a). Because of the reciprocal relation between rand
Uin Eq. (3), an increase in rreduces U[see Fig. 5(b)].
This enables a smaller Uspacing between consecutive
electron transfer peaks, leading to a lower required single
electrode voltage V,max for a targeted density of charge
storage [see Fig. 5(a)]. The maximum density of charge
that can be stored is determined by the number of coun-
terions that can packed in to maintain charge neutrality
[see Fig. 2(a)][34]. Ideally, Ushould also be higher than
the thermal energy of about kBT—though this is not a
strict necessity. Hence, when tuning the maximum volt-
age (V,max), a Uvalue within the range of 0.025 to 0.1
eV is preferable when working to maximize the voltage at
Required Voltage (V)
20 40 60 80 100
1 nm
2 nm
3 nm
4 nm
5 nm
6 nm
7 nm
8 nm
9 nm
10 nm
20 40 60 80 100
20 40 60 80 100
FIG. 5. (a) Achieving a target single electrode voltage for
adding a target density of electrons via quantized capacitance
by tuning the dielectric response and nanoparticle radius. (b)
The change in parameter Uin response to tuning the dielectric
response and nanoparticle radius as described by Eq. (3).(c)
The change in reorganization energy λwith various dielectric
responses and nanoparticle radii. Note that (a) is computed with
a target density of electrons of 4 q/nm2.
a high density of electrons storage [10]. Comparing Figs.
5(a) and 5(b) we see that disks with a radii in the range
of 3–4 nm within a dielectric medium characterized by r
between 40–60 would serve well. This would then provide
an operating voltage contribution of about 2.5 V per termi-
nal (at σe=4q/nm2) for a total of about 5 V. However,
these estimates are derived classically and a more refined
search should consider atomic scale interactions, as will be
discussed shortly.
The reorganization energy λof a given particle is also
dependent on the radius and dielectric medium of such
a nanodisk. Its outer-sphere contribution can be directly
computed from Eq. (4) and is plotted as a function of rfor
various nanodisk radii in Fig. 5(c). Here we are assuming
that op =2. The reorganization energy is important in that
it enables a smooth pseudocapacitive current by providing
sufficient overlap between many overlapping redox peaks,
as governed by Eqs. (3) through (7).Ifλis too small rela-
tive to U, the ability of this mechanism to provide a smooth
capacitive voltammetric profile, such as that in Fig. 4(a),
can become hampered, as discussed in Ref. [10]. Hence,
ideally, one would like to maintain λU. From the results
in Fig. 5(c), this should also be satisfied by nanodisks with
radii of 3–4 nm within a dielectric medium characterized
by rbetween 40–60. The reorganization energy is also
an important contributing factor in the power density per-
formance of the energy storage mechanism envisioned in
Fig. 2(a), which we will discuss in Sec. III.
Lastly, it is important to recognize that the classical esti-
mates in Fig. 5exclude: (1) the screening response [6872]
and space charge polarization [73,74] of counterions; and
(2) the inner-sphere reorganization response of the solvent
electrolyte molecules. Hence, the results in Fig. 5serve
only as an approximate physical estimate. Detailed atom-
istic calculations are needed to more accurately compute
the combined counterion and molecular-scale contribu-
tions to the charging and reorganization energies in a given
supporting medium [7577]. However, it has been exper-
imentally demonstrated that a charging energy response
should be present at the nanoscale in such a system, as
demonstrated in Ref. [78] as well as other works [17,29,
31,79]. Hence, the general physical arguments presented
should hold—above and beyond system specific details.
Overall, the results in this subsection are intended to con-
vey the importance of tailoring both the dielectric medium
and nanodisk dimensionality. Both should be tuned to
attain a target electron storage density (per disk) at a given
storage voltage via the quantized capacitance mechanism.
D. Electrolyte considerations in quantized capacitance
As mentioned briefly in Sec. II B, charge storage via
quantized capacitance occurs over two electrodes [see
Fig. 2(a)]. One electrode serves as the negative () termi-
nal by gaining electrons during charging, while the other
serves as the positive (+) terminal by giving up electrons
during charging. Since opposite charges are stored on both
electrodes, the terminals will be biased in opposite direc-
tions with respect to their fully discharged configuration.
Hence, the overall cell potential in Eq. (1) is the addition
of the biases on both electrodes as described by
Here, |V+,max|is the maximum applied bias dropping at
the positive terminal and |V,max|is that of the negative
terminal (see Fig. 4). In the simple case of symmetric elec-
trodes, the biases on both electrodes will be approximately
equal such that |V+,max|≈|V,max|≈Vd/2, as shown in
Fig. 6(a). However, it is also possible that the total volt-
age (Vd)inEq.(8) may be split asymmetrically across two
terminals (|V+,max| =|V,max|) with each storing an equal
amount of charge. Following from the results of Fig. 5, this
YEE WEI FOONG et al. PRX ENERGY 1, 013007 (2022)
asymmetric splitting may occur when the nanodisk radius
and/or the supporting medium dielectric response is not the
same at both electrodes. For example, suppose that the pos-
itive (+) terminal has particles twice (2 times) the radius
than those on the negative () terminal. Then, keeping all
other system parameters fixed, the positive terminal (+)
will only require |V+,max|=|V,max |/2 to store the same
density of electrons (see Fig. 5). Under this scenario, 2Vd/3
would drop across the negative () terminal and Vd/3
would drop across the positive (+) terminal. More gener-
ally, the manner of voltage splitting matters because it can
be utilized to maximize energy storage within electrolyte
stability constraints, as we now explore.
This voltage splitting arrangement relates directly to
how positive and negative charges can be stored in the
manner proposed in Fig. 2(a). It is well known that carbon
nanostructures excel at storing electrons [2325]. Indeed,
as mentioned in Sec. II, an electron storage density of
about 4 q/nm2can be routinely achieved. However, the
propensity for electron removal from carbon nanostruc-
tures is much more challenging [80,81]. For example,
while one can place up to about seven electrons on a
C60 molecule, only up to three electrons can typically
be removed [25,78,80,81]. Whether one is considering
fullerenes or another carbon nanostructure, this difficultly
arises when the removal of electrons from the electrolyte
(breakdown) occurs at an earlier potential than the removal
of further electrons from the intended (carbon) nanostruc-
ture. On the other hand, it is possible to attain stability upon
removal of high densities of electrons in bulk graphitic
systems. For example, in BrC8graphite sheets give up
to about 4.8 q/nm2[82]. More recently, similar success
has been found in FeCl3-doped graphene [83]. Compara-
tively, in C60 the ratio of electrons that can be removed
(even in the presence of counterions) is around one for
every 20 atoms, versus one for every eight atoms in BrC8.
The challenge, of course, is how to achieve the excel-
lent electron-electron removal capabilities of graphite or
graphene in a smaller nanostructure where the V+,max stor-
age voltage can be tuned following Sec. II C—prior to
reaching electrolyte breakdown.
The contrasting ability of graphite to give up more elec-
trons than C60 is due to the energies at which electron
removal can be accessed. Since electrons are well delocal-
ized in graphene or graphite, the quantization and electron-
electron interaction energetic costs associated with elec-
tron removal (or addition) are much less than in C60 [10,53,
54,69,84]. Consider the comparative electronic structure
plots for graphene and C60 in Fig. 6(a). Here we can see
that the HOMO level of C60 lies about 6 eV below vacuum,
while the charge neutrality point (Dirac cone) of graphite
or graphene lies at about 5 eV below vacuum as calculated
from first principles—see the Supplemental Material [56]
and also Refs. [8589]. Now, in Fig. 6(a) we have super-
imposed in green the stability window of a hypothetical
electrolyte ranging from 2to7 eV below vacuum. This
range is chosen for its potential to be realized through
electrolyte engineering methods [58,9092]; it has also
been placed above band structure plots for graphene and
BrC8decorated graphene [9395]. Clearly, the removal of
all such electrons from single-layer BrC8lies within the
stability window of this electrolyte [96]. However, assum-
ing a charging energy of U=0.3 eV after the removal
of about three electrons from C60 we find that the stabil-
ity limit of the electrolyte is reached [17]. Note that the
HOMO level of C60 is sixfold degenerate. Hence, beyond
the HOMO-LUMO gap only the charging energy con-
tributes to the cost of electron removal of the first six
electrons in C60 [17,78]. The point here is that it is the
dimensionality of graphite (graphene), being a bulk 2D
(3D) material, that allows a very high density of electrons
to be removed with minimal energetic cost (U). This can
be seen in the far right single-layer BrC8band structure in
Fig. 6(a), where the Fermi (Ef) level lies well above any
electrolyte stability considerations. However, in C60 the
energetic cost of electron removal (U) is much higher and
so much less can be removed at the same potential (or any
potential prior to electrolyte breakdown). Thus, the dimen-
sionality of a nanostructure directly impacts on the density
of charge it can store prior to reaching the voltage limit at
which electrolyte breakdown occurs. Conversely, one can
maximize the voltage (prior to electrolyte breakdown) at
which a target electron density is removed from graphitic
nanostructures (e.g., disks) by tuning their dimensionality
(see Fig. 4). Intriguingly, the ability to store more electrons
within a fixed bias range with increasing radius (and to do
so with high amphoteric propensity) has been experimen-
tally demonstrated in graphene nanoparticles via quantized
capacitance [29].
To overcome these electrolyte stability issues, which
compete with the electron storage and removal, we briefly
explore two possible engineering avenues for the system
proposed in Fig. 2(a). First, one can attempt to engineer an
electrolyte that has a large stability window that is directly
symmetric about the Dirac cone of graphene or graphite
(at about 5 eV), as shown in Fig. 6(a). In this manner,
the charging levels of smaller graphitic nanodisks will also
align about the Dirac cone, allowing for the use of symmet-
rically designed positive and negative terminals with the
same disk size, which in turn maximize the storage voltage
for the target electron density, as shown in Fig. 6(a).The
framework for relating the single-particle ionization and
affinity energies (such as the “Dirac cone”) in Fig. 6(a) to
voltammetric spectra, such as breakdown voltages relative
to a reference electrode, can be found in Refs. [17,98]—see
also the citations therein. Promising electrolytes, which
might be stable for a wide region about the “Dirac cone”
in graphene or graphite could perhaps include acetonitrile
and/or sulfolane [92]. Second, one can independently tune
the nanodisk dimensions on each electrode to fit a given
(a) (b)
+6 +5 +4 +3 +2 +1
+6 +5 +4 +3 +2 +1
–1 –2 –3 –4 –5 –6
FIG. 6. (a) Comparative electronic structure plots against an electrolyte stability window (green) situated between 2and7eV
below vacuum. The left panel shows the electronic structure of C60, the middle panel that of graphene, and the right panel that of BrC8
decorated graphene. A dashed line provides the alignment of all electronic structure plots to the Fermi energy (Ef) of graphene. The
Fermi energy of single-layer BrC8lies below that of graphene, due to the loss of electrons to Br atoms, but above the stability limit
of the electrolyte window. (b) A schematic depicting the position of the open circuit potential (VOC) within the electrolyte stability
window (shaded in light blue) dictates the electrode potentials (|Vanode|and |Vcathode |) and the maximum cell potential Vmax for a system
with symmetric electrodes. This schematic assumes the simple case that VOC lies at the center of the electrolyte stability window with
the equally distributed current peaks (shaded in gray) for electron removal at the positive terminal and electron addition at the negative
terminal for different charge states on the graphitic nanodisks during a charging process. Here, the equal peak distribution (U+=U)
on each terminal is achieved by nanodisks of similar radius on both sides. The inset shows the definitions of anode and cathode for a
supercapacitor cell under the charging process [97]. (c) A schematic for a more frequently observed VOC that lies closer to the limits
for the positive electrode. To avoid wasting the wider potential window on the negative electrode, the nanodisks can be engineered to
have a smaller radius for a larger Uas compared to U+.
electrolyte stability window alignment. This scenario is
shown in Fig. 6(b), where the Dirac cone of graphite or
graphene lies closer to the positive breakdown potential
of the electrolyte than to its negative breakdown potential.
In this case, the nanodisk radius on the positive electrode
should be larger than that on the negative electrode, so
as to store the same density of charge but a lower poten-
tial relative to the fully discharged configuration about the
Dirac cone energy [see Fig. 6(b)]. Conversely, the nan-
odisks on the negative electrode can be made smaller to
provide a larger voltage window for storing the same den-
sity of electrons per disk by realizing a higher charging
energy cost (see Fig. 4)[29,99]. Additionally, one may
overcome instability via distinct electrolytes on each termi-
nal, as discussed in the Supplemental Material [56]. Other
approaches to overcome electrolyte stability certainly exist
and are left for future work [58,90,91]. Given the the-
oretical challenges associated with predicting electrolyte
breakdown voltages, a detailed experimental investigation
of quantized capacitance charging and electrolyte stability
is needed to further efforts in this direction [78].
The power density of conventional EDL-based super-
capacitors is essentially determined by the diffusion of
counterions in the charging and discharging processes
[2,100102]. When the ionic diffusion constant reaches
around 1010 to 109m2/s[103105], these EDL-based
systems can achieve high power densities of approximately
104W/L (see Fig. 1). To match the power density of purely
EDL-based supercapacitors as suggested by Fig. 1,a
YEE WEI FOONG et al. PRX ENERGY 1, 013007 (2022)
pseudocapacitive mechanism must exhibit fast and
reversible redox activity. While the reversibility of a redox
system utilizing quantized capacitance is very much deter-
mined by the design considerations discussed in Sec. II D,
the rate of redox activity is determined by the speed at
which electrons can transfer into and out of nanoparticles,
as illustrated in Fig. 2(a). Hence, there are two mechanisms
that determine the power density of quantized capacitance
as an energy storage medium: (1) counterion diffusion;
(2) electron transfer and diffusion. Going forward we will
assume that the ionic diffusion engineering issues are
similar to either those of conventional EDL-based super-
capacitors or organic radical batteries—leaving a detailed
atomistic analysis to future work [106108]. Instead, we
focus on the manner in which the electron transfer should
also be engineered to maximize the power density of this
mechanism (see Fig. 1).
In order for quantized capacitance to persist, particles
must be separated by a reasonable tunneling barrier. This
is necessary to promote Coulombic interactions between
electrons on a nanoparticle and thereby arrive at a “quan-
tized” value of Uas described by Eq. (3) [10,17]. By tuning
Uwe are able to engineer the storage voltage for a target
electron density σe, as discussed in Sec. II. However, this
essential tunneling process cannot be so slow as to render
the power density impractical (see Fig. 1). When consid-
ering the overall proposed mechanism in Fig. 2(a), the
key limiting factor is the rate at which electrons are trans-
ferred between individual particles via tunneling. In the
Gerischer-Hopfield description of quantized capacitance,
this interparticle electron tunneling (transfer) rate can be
approximated as [10,17,63,6567,109111]
kip =4π2|Mip|2
where |Mip|is the electronic coupling between particles, h
is Planck’s constant, and λc=2λis the classical Marcus-
Hush reorganization energy. In Fig. 7(a) we can see that kip
depends primarily on both |Mip|and λc. The key assump-
tion in Eq. (9) is that the electrons are weakly coupled such
that the transfer mechanism is an outer-sphere (tunneling)
process [62]. This intersite electron transfer mechanism is
essentially the same as that present in redox-polymer bat-
teries [60,112114]. Now if we further assume that the
nanodisks in Fig. 2(a) are uniformly spaced, such that the
electronic coupling |Mip|to all nearest-neighbor particles
is approximately the same, then the diffusion of electrons
in this system can be approximately written as
2dkip, (10)
0.1 0.2 0.3 0.4 0.5
Mip (eV)
1.5 2.0 2.5 3.0
Mip (eV)
0.5 1.0
Barrier Width (nm)
Mip (eV)
0.1 0.2 0.3 0.4 0.5
FIG. 7. (a) The |Mip|and λcto achieve a specific electron
transfer rate kip, ranging from 103to 100 s1. (b) The effect
of barrier width on the required Mip to efficiently store the elec-
trons at various barrier heights Vb. The value of (c) Deat 300 K
as a function of |Mip|and λcfor a 2.5 eV tunneling barrier.
where d=3 is the dimensionality of the hopping process
and lis the hopping distance [115]. This then allows us
to rapidly arrive at an approximate understanding as to
how electron diffusion should be coengineered with ionic
diffusion in this system.
Assuming that the temperature is held fixed at about 300
K, the interparticle electronic coupling (|Mip|) and classical
Marcus-Hush reorganization energy (λc=2λ) primarily
dominate the diffusion of electrons via Eqs. (9) and (10).
To first order, the electronic coupling is further dependent
upon the tunneling barrier width Wand height Vbbetween
the particles in the manner of
|Mip|≈2|Vb|exp W2m|Vb|
, (11)
which is related to the Gamow tunneling expression [116,
117]—mis the electron mass and =h/2π. In Fig. 7(b) it
can be seen that both the tunneling barrier height (Vb)and
width (W) exponentially impact |Mip|and thereby the elec-
tron diffusion rate (De). The overall magnitude of Deas a
function of both |Mip|and λcis plotted in Fig. 7(c). Here it
can be seen that to achieve a target electron diffusion con-
stant in the range 1010–109m2/s, comparable to high
performance ionic diffusion, one needs to attain reorgani-
zation energies (λc) in the range 0.15–0.25 eV and elec-
tronic coupling strengths (|Mip|) in the range 102–104
eV. Returning to Fig. 5(c) we can see that reorganization
energies in this range should be easily achievable with
r>20. Note that these estimates do not include inner-
sphere contributions to the reorganization energy, which
should raise their estimate magnitudes further and impact
De. However, the electronic coupling range estimated may
be more difficult to achieve from a practical engineering
Based on the estimates in Fig. 7(b), a tunneling barrier
with a height of Vb=2–3 eV and a width of about 1.2
nm would work best to provide target electron diffusion
values in the range 1010–109m2/s [see also Fig. 7(c)].
In this system the tunneling barrier is determined by the
electrolyte stability window, that is the offset between the
liquid electrolyte HOMO and LUMO levels from the nan-
odisk levels in Eq. (7), and can be manageably engineered
[118,119]. The requirement for maintaining an interdisk
separation of about 1.2 nm is much more difficult to imple-
ment, as it is directly coupled to the volumetric packing of
such nanodisks, as discussed earlier in the context of Fig. 3.
This somewhat stringent spacing criteria, which excludes
any randomness in packing, could likely be alleviated
by introducing electron shuttling sites such as buckyballs
[120122]. Such species could act as intermediate trans-
fer centres to carry electrons between nanodisks that are
separated by more than about 1.2 nm, due to packing
randomness, and thereby prevent the effective electronic
coupling between nanodisks from becoming too low. Inter-
estingly, an interdisk spacing corresponding to about 1.2
nm could also aid the diffusion of counterions by pos-
sibly removing the traditional solvation shell limitations
[123125]. From this perspective, our theoretical estimates
for optimum packing to promote electron diffusion may
also similarly aid counterion diffusion. Here, it is impor-
tant to realize that both electron diffusion and ion diffusion
can impact the power density of a pseudocapacitive battery
operating via quantized capacitance. This aligns with the
findings reported in Refs. [114,126], where the concerted
diffusion of ions and electrons are shown to impact energy
performance. However, a conclusive exploration of these
issues requires detailed atomistic calculations coupled with
careful experimentation [106,108,127130].
Before concluding, it is helpful to consider existing
technological reference points through which this energy
storage design might be better understood. As briefly men-
tioned in Sec. I, redox-polymer batteries are perhaps the
most relevant practical analogue to the system explored
herein (see Fig. 2)[22,131]. In both designs redox centres
are dispersed in a supporting medium, with operation facil-
itated by the classical diffusion of counterions and the tun-
neling (outer-sphere transfer) diffusion of electrons [126].
Unlike redox-polymer batteries, quantized capacitance is
capable of producing multiple redox reactions on a single
site, in such a manner that the Faradaic current is able to
mimic a pseudocapacitive response, as shown in Fig. 4[10,
17,54]. In a sense one could think of the proposed mecha-
nism as a “pseudocapacitive battery”—though quantized
capacitance is not limited to carbon-based nanoparticles
[1116]. Clearly, the physical distinction between a pseu-
docapacitive Faradaic mechanism and that of a battery
begins to blur in this domain.
Overall, reliable synthesis likely remains the major
obstacle to prototyping quantized capacitance as a stor-
age mechanism—though electrolyte stability should not be
ignored, as discussed in Sec. II D. Firstly, one should be
able to produce low variability in the synthesized dimen-
sions of nanoparticles. This is needed to ensure that the
nanoparticle charging energy is tailored to store a desired
electron density at or near a target voltage, as discussed
in Sec. II C. Though certainly some degree of tolerance
in synthesis size within an ensemble should be accept-
able [16,32]. Secondly, the requirement that they be singly
packed and separated by reasonably well-controlled tun-
neling barriers could be difficult to fabricate—particularly
at a large scale. Nevertheless though, cumulative engineer-
ing development quantum dots have been integrated into
large-scale fabrication processes [132134]. With impetus,
similar engineering advances might make the quantized
capacitance scheme in Fig. 2(a) more practically tangible.
For example, it has recently been shown that the inter-
planar distance of graphene oxide can be finely tuned
[23,24], thereby demonstrating the possibility to precisely
control the interparticle distance in supercapacitor systems.
Additionally, carbon nanotubes of uniform structure and
dimensionality can now be routinely fabricated [135137].
Similar approaches might be applied to make this theoret-
ically benchmarked system a realizable “pseudocapacitive
battery” technology.
There is also a possibility of combining the proposed
quantized capacitance setup within other emerging EDL
technologies. For example, rather than utilizing the single
YEE WEI FOONG et al. PRX ENERGY 1, 013007 (2022)
electrode setup in Fig. 2(a), a matrix of nanodisks could
be imbedded between stacked MXenes sheets [138140].
The nanodisk matrix would add a Faradaic storage compo-
nent to such a supercapacitor system. Likewise, vertically
aligned MXene layers would also aid fast ionic diffusion.
In this manner, two such technologies might complement
each other to improve overall performance (see the Sup-
plemental Material [56]). Similar hybrid approaches could
be applied with other supercapacitor systems [38,40,141].
Even a slurry-type redox flow battery utilizing quantized
capacitance to access multiple redox states is plausible
Returning to the Ragone plot in Fig. 1, all of the theo-
retical analysis conducted herein suggests that quantized
capacitance might be engineered to combine the power
performance of supercapacitors with the energy density
of battery systems. Theoretically, quantized capacitance
could yield an energy density of 100 Wh/L combined
with a power density of 104W/L. This would be a tangi-
ble benchmark worth pursuing in the development of this
mechanism [21,142,143], and further exploring system-
specific engineering parameters in more detailed studies.
However, in our analysis we have omitted the gravimetric
energy density, instead focusing on the volumetric den-
sity. This is a key point to consider and relies entirely on
the materials utilized to construct such a system, with the
number of chemical permutations being vast. Optimisti-
cally, a primarily carbon-based system [as illustrated in
Fig. 2(a)] would offer a high gravimetric density. How-
ever, the choice of counterions also plays a critical role in
determining the gravimetric density [144146]. Going for-
ward, close collaboration between theory and experiment
are needed to resolve and analyze specific systems with the
aim of fabricating such a “pseudocapacitive battery” capa-
ble of the theoretical limits outlined herein. Future work
should focus on experimentally benchmarking specific
systems through state-of-the-art characterization coupled
with sophisticated atomistic calculations [39,147,148].
All authors gratefully acknowledge financial support
from the Natural Sciences & Engineering Research
Council of Canada. Computational resources were pro-
vided by the Canadian Foundation for Innovation, Cal-
culQuebec and Compute-Canada. The authors would like
to acknowledge financial support from EPSRC Grant
No. EP/R034540/1 for the JSPS-EPSRC-McGill Univer-
sity collaboration on “Defect Functionalized Sustainable
Energy Materials: From Design to Devices Application”
and UCL for travel support.
Y.W.F. conducted the analysis, benchmarking, and elec-
trolyte optimization content. K.H.B. developed the for-
malism and energy storage concept. C.A.H. contributed
the electron removal and electrolyte stability concepts
and analysis associated with carbon nanostructures. The
first-principles calculations were conducted by J.S. and
S.Y. under the supervision of K.H.B. All of the authors
discussed the results and participated in composing the
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