Subscriber access provided by LAKEHEAD UNIV
Article
Charging Dynamics and Optimization of Nano-Porous Supercapacitors Svyatoslav Kondrat, and Alexei A. Kornyshev J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp400558y • Publication Date (Web): 07 May 2013 Downloaded from http://pubs.acs.org on June 5, 2013
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
The Journal of Physical Chemistry
Charging Dynamics and Optimization of Nano-Porous Supercapacitors S. Kondrat and A. Kornyshev Department of Chemistry, Faculty of Natural Sciences, Imperial College London, SW7 2AZ, UK (Dated: March 28, 2013) An improved mean-field model used earlier (J. Phys.: Condens. Matter, 23, 022201, 2011) to explain the anomalous increase of capacitance in nano-porous supercapacitors is extended to the study of charging dynamics. We find that charging of initially empty pores (non-wettable by ionic liquids) proceeds in a front-like way, while charging of filled (wettable) pores is diffusive and appears to be initially slower but asymptotically faster in the limit of large charging times. This suggests the use of porous materials wettable by ionic liquids to maximize the power density for thick electrodes. We also discuss two-step complementary optimization of porous electrodes for supercapacitors. In a first step, the optimal pore width is chosen to maximize the stored energy density; in a second step, the optimal pore depth/length (that is, electrode’s thickness) is chosen to satisfy the requirement on charging times. In addition, the use of ‘nano-porous channels’ in a ‘multi-layered’ configuration is suggested to decrease the volume and to increase the capacitance, stored energy and power density of a supercapacitor. Keywords: Ionic liquids, ionic diffusion, electric double layer capacitors, ultracapacitors
I.
INTRODUCTION
Electric double layer (EDL) capacitors literally ‘fill the gap’ between dielectric capacitors and conventional batteries1 . Also known as ultra- or supercapacitors, they can store energy thousands of times greater than a typical dielectric capacitor, and are able to supply the stored energy within seconds, hundreds of times faster than a conventional battery. In addition, supercapacitors show excellent durability, being able to withstand up to 106 charge/discharge cycles. It is due to these properties that supercapacitors have attracted such enormous attention in industry and academia. A typical EDL capacitor consists of two porous carbon electrodes immersed in an electrolyte medium. The energy is stored in the electrical double layers formed at the interface between the porous electrodes and electrolyte. Optimization of porous carbons, aimed at increasing the stored energy and power density, is thus of a great technological importance. While there is some progress in understanding of the energy storage in porous carbons2–7 , studies of charging dynamics are rather scarce. Most, if not all, theoretical studies of pore-charging dynamics have been limited to dilute electrolyte solutions and/or to charging of pores that are wide in comparison to the thickness of the electric double layer in the electrolyte, which is typically a few nanometers (see Refs. 8– 10 and references therein). A popular model, frequently used by engineers and other researchers, is de Levie’s transmission line11 . It maps a pore onto an ‘equivalent circuit’ consisting of ‘capacitors’, due to the double layers forming at the pore walls, and ‘resistors’, describing the resistance that the ionic liquid faces when moving along the pore. It can be shown12 that the transmission line type model is the zero-order, linear approximation of the more general Nernst-Planck-Poisson equation. Nevertheless, it seems that both the transmission line and the full Nernst-Planck-Poisson approach predict diffusive charg-
ing, with the accumulated charge growing as the square root of time8 . To the best of our knowledge, there appears to be no Molecular Dynamics simulations studies of the pore charging dynamics. Apart from possible lack of interest to this problem in the past, the main reason for this seems to be a high requirement for the computational resources. Indeed, dynamics of ionic liquids is known to be very slow, as reflected in their high viscosity, thus large computational time is needed to obtain reliable results. In addition, realistic pores can be hundreds of nanometers to millimeters long, that again appeals for high computational demand. It has been shown that the maximum in capacitance2 and in stored energy density6 occurs for nano-sized pores, specifically for pores close in size (L) to the diameter (d) of ionic liquid molecules. This work focuses on the charging dynamics of such narrow nanopores. For this purpose, we extend the model4 that we used earlier to explain the ‘anomalous’ increase of capacitance2 for L ≈ d (section II). Due to some important subtleties, we distinguish and study separately the case when the pores are non-wettable (section III A) and wettable (section III B) by ionic liquids at zero voltage. Based on this study, we also outline possible routes to the optimization of porous supercapacitors (section IV). Finally, in section V we summarize and discuss criticaly our findings.
II.
MODEL AND METHOD
We consider a porous electrode immersed in an electrolyte medium. We are interested in electrode’s charging after its potential is raised with respect to the bulk of the electrolyte (or another electrode). Such charging process can be conveniently split into three stages. In the first stage, the non-zero charge density is established at the outer surface of an electrode, in response to the applied voltage; as if it were a flat, solid electrode. This process
ACS Paragon Plus Environment
The Journal of Physical Chemistry
Page 2 of 9 2
I (f inal)
x=0
ρ±
L
II
(eq)
ρ± (V = 0)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
x
hence we neglect the entering and closing/exit of a pore (see, however, section III C). We also assume that the pore is very narrow and consists of a single ionic liquid layer. Such model was used in Ref. 4 showing a qualitative agreement with simulations and experiments5 . The free energy of our system is given by Z XZ F [ρ± ] = dR1 dR2 ρα (R1 )Uαβ (R1 , R2 )ρβ (R2 ) α,β
III
FIG. 1. Charging of a narrow nanopore undergoes three stages. In stage I, the non-zero charge density is established at the outer surface of the electrode including the pore entrances. In stage II, the charge density propagates along the major part of the pore (of width L). Finally, the charging is being completed during stage III when the equilibrium ion densities are established along the whole pore, including its closing. In this work we focus mainly on stage II, denoted by a dashed rectangular area (see sections III A and III B), however, we shall discuss stage III for open pores in some details as well (see section III C). To study pore-charging dynamics, the equilibrium ion densities at zero voltage are set inside the (eq) (initial) = ρ± (V = 0), pore at t = 0, i.e. ρ± (x > 0, t = 0) = ρ± while certain, not necessary equilibrium densities are kept at (f inal) (for the pore ‘entrance’ at all times, ρ(x ≤ 0, t) = ρ± details see sections II B and III).
has been widely studied in the literature (see Refs. 12, 13 and references therein) and we do not discuss it here. In the second stage, the charge density ‘propagates’ along the pores, in response to the gradient of the charge density and chemical potentials created at the pore entrance. We assume here that the time scale of the first stage is faster than the time scale of the second stage, so that these two stages can be considered independently. The third stage starts when the propagating ‘wave-front’ hits the closing of the pore. Evidently the first stage lasts for the whole duration of the charging process, as through it the counter-ions are supplied for and the co-ions are ‘consumed’ during the second and third stages. As the pore-depth of a typical porous electrode, i.e. electrode’s thickness14 , is of order of hundreds micrometers1 , the second stage is most time-consuming and therefore deserves a special consideration. As mentioned in the introduction, charging of pores wide comparing to the width of the electric double layer has been studied in the literature, and we focus here specifically on charging of nano-sized pores.
A.
Model
We consider a single slit-like metallic pore; this type of model porous electrode has been used by many authors to study capacitive properties of nanoporous supercapacitors4,5,15,16 . Following these works, we shall assume that a pore is infinitely extended in the two lateral directions,
+
X α
hα
Z
dR ρα (R) + F0 [ρ± ], (1)
where R = (x, y) are lateral coordinates, ρα={±} (R) are two-dimensional ion densities, and hα (V ) = µα + qα V + δEα − kB T Zα2 ln(2)LB /L.
(2)
Here µα is the chemical potential and qα = eZα the ion charge, where Zα is the ion valency and e the elementary charge; δEα is the so-called ‘resolvation’ energy4 , L is the pore width and V the voltage (electrostatic potential at the pore walls); LB = e2 /εp kB T (in Gaussian units) is the Bjerrum length17 , where εp is the dielectric constant inside the pore (assumed pore-width independent18 ), kB is the Boltzmann constant and T temperature. The last term in Eq. (2) comes from the ion self energy (see Ref. 4 for details). The first term in Eq. (1) is the electrostatic energy. We use the following approximation for Uαβ to account for short-range correlations: Uαβ (R1 , R2 ) = gαβ (R1 , R2 ) Φαβ (R = |R1 − R2 |) , (3) where19 Φαβ (R) =
∞ 4qα qβ X 2 sin (πn/2) K0 πnR L εp L n=1
(4)
is the electrostatic interaction potential between αβ ions confined by two metal walls, where Kn is the modified Bessel function of the second kind of order n. We stress that Φαβ preserves a constant potential at the pore walls that belong to the same electrode. It is also important to note that Eq. (4) is derived for infinitely extended metal walls, hence it is not valid near pore’s entrance and closing. The case gαβ = 1 corresponds to the standard meanfield approximation; it neglects the short-range correlations and, as a result, does not capture the anomalous capacitance increase4 . Here we choose gαβ (R, R′ ) = θ (|R − R′ | − bαβ ). In Ref. 4 we used bαβ = Rc (ρ) = (π/ρ)1/2 , where ρ = ρ+ + ρ− is the total ion density. In this work, however, we assume a constant bαβ for simplicity; although this assumption does not lead to any phase transitions20 discussed in Ref. 4, it retains an anomalous capacitance increase (c.f. Fig. 3(a)). R Finally, the third term in Eq. (1), F0 [ρ± ] = f0 (ρ± )dR, is the ‘reference’ free energy in the local density approximation, where for f0 (ρ± ) we adopt the Borukhov-Andelman-Orland form21 to account for excluded volume effects (see Ref. 4 for details).
ACS Paragon Plus Environment
Page 3 of 9
The Journal of Physical Chemistry
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
B.
Dynamics
To study dynamics we use an approach reminiscent of a dynamical density functional22–27 , that is, we assume that the evolution of the ion density is governed by the continuity equation ∂ρ± δF = −∇J (±) , = Γ ∇ ρ± ∇ ∂t δρ±
charge density, c/ρmax
3 1 0.8 0.4 0.2 0
(5)
where J (±) = −Dρ± ∇ (βδF/δρ± ) is the current of ± ions, β −1 = kB T , Γ is a two-dimensional phenomenological mobility parameter, D = Γ/β is the ‘diffusion constant’ setting up the time scale, τ = d2 /D, where d ≡ d± is the ion diameter. A similar approach has also been taken in recent studies of charging of flat12,13 and wide porous8,9 electrodes. Equations (5) and (1) form a system of non-linear, nonlocal integro-differential equations and must be supplemented by initial and boundary conditions. It is important to stress here that h(V ) does not enter dynamic equations (5), as it drops down due to the presence of gradients in Eq. (5). Hence the dynamic equations define the way the system develops in time between the two (thermodynamic) states, while the states are specified by the initial and boundary conditions. In what follows we set the initial ion densities inside the pore to the equilibrium densities corresponding to zero (initial) (eq) = ρ± (x > 0, t = 0) = ρ± (V = 0). voltage, i.e., ρ± (eq) We also choose ρ± (x = ∞) = ρ± (V = 0) at all times (except of section III C); this suffices to study the second stage of the charging process (see Fig. 1), and is in agreement with our assumption of an ‘infinitely’ elongated pore. Alternatively, one could choose ∂x ρ± (x = ∞) = 0, similarly to the works of De Levie and others. It is clear, however, that these two boundary conditions are equivalent and shall lead to the same results, as long as stage II of the charging process is concerned. Now the pore charging is specified by setting certain ion densities at the ‘pore entrance’, ρ± (x < 0, t ≥ (f inal) 0) = ρ± , which we assume the same for x < 0 and which correspond to the ‘final’ state (see below). In other words, we assume a sharp ‘interface’ between (initial) (eq) (f inal) ρ± = ρ± (V = 0) and ρ± at t = 0. As we will see, this bounds us to use different strategies to study charging of initially empty28 and filled pores. We dis(f inal) cuss the choice of ρ± in subsections III A and III B, respectively. We solve Eqs. (5) supplemented by the appropriate boundary and initial conditions numerically using the GNU Scientific Library29 . For the sake of simplicity we shall also assume the local density approximation in the first term of Eq. (1). We have checked for a few values of the model parameters30 that the local density approximation does not lead to any significantly qualitative changes.
t/τ = 1 t/τ = 2
0.6
0
5
10 15 x, nm
20
FIG. 2. Front-like propagation of the charge into an empty nanopore after a voltage jump. The charge density c = ρ+ − ρ− has been calculated using Eqs. (1) and (5). The Bjerrum length LB = 10 nm, ‘resolvation’ energy δE± = 15kB T , ion diameter d = 0.7 nm and the pore width L = 0.8 nm. The densities inside the pore at t = 0 are set to the equilibrium densities corresponding to zero voltage and are close to zero. The densities at x = 0 are set to the equilibrium densities at V = 1.3 V (c.f. Fig. 3) and are close to the maximum density allowed within our theory.
III.
PORE CHARGING DYNAMICS
We are primarily interested in two quantities. One quantity is the ion densities along the pore at different times, i.e., we want to see how the initially assumed charging ‘front’ develops in time. A quantity of more practical interest is the total charge Q accumulated in the pore as a function of time. It is also interesting to look at the effective charging depth, Hef f , which we define as follows31 Z ∞ Hef f (t) = q0−1 Q(t) = q0−1 ec(x, t) dx, (6) 0
where c = ρ+ − ρ− is the volumetric charge density, q0 = ec(x = 0) is the ‘final’ charge density (for choice of q0 see below), and e is the elementary charge. The Hef f can be interpreted as the charging depth of an ‘equivalent’ pore whose charging proceeds in a strictly front-like way. In the following we discuss separately charging of the initially empty and filled pore.
A.
Charging of empty pores
To study charging dynamics of empty (or nearly empty) pores, we proceed as outlined in section II B. We set the ion densities at x = 0 to the equilibrium ion densities corresponding to voltage V > 0, where x = 0 is put some distance away from the actual pore entrance (see Fig. 1), so that its effect on the ion interaction potential, Eq. (4), can be ignored. In doing so we assume that charging proceeds in a front-like way, and approximate a moving front at t = 0 by a sharp ‘interface’ between the two equilibrium states corresponding to V 6= 0 and V = 0. This assumption is confirmed a posteriori by
ACS Paragon Plus Environment
The Journal of Physical Chemistry
Page 4 of 9
10 L = 0.9 nm L = 0.8 nm 0
0
0.5 voltage, V
1
(b) full charging
1
0
Hef f , nm
charge, nC/m
Fig. 3(b)). The difference is not so significant, however, if we compare the effective charging depth (see inset in Fig. 3(b)). This suggests that wider pores charge faster mainly due to the higher final equilibrium charge density (set at x = 0), leading to higher charge-density gradients created at the interface, and hence stronger driving forces. It is interesting to look at voltages below saturation to find charging of smaller pores significantly faster (Fig. 3(c)). This also seems to result from larger density gradients created at the interface, being a direct consequence of the anomalous capacitance increase.
20 (a)
2
0
2
8 4
B.
0
time, t/τ
2
3
0
8 4 0
Charging of filled pores
time
(c)
1
0
1
Hef f ,nm
charge, nC/m
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
total charge, µC/cm2
4
voltage
0.5 1 voltage, V
t/τ = 2 1.5
FIG. 3. Charging of initially empty pores. (a) Equilibrium (eq) total charge accumulated in the pore per surface area, Qtot , as a function of applied voltage. The narrower pore stores less charge (and hence less energy) when fully charged but its (eq) corresponding differential capacitance, C(V ) = dQtot /dV , is higher, in accord with the anomalous capacitance increase2 . (b) Accumulated charge per unit length, Q, and effective charging depth, Hef f (see Eq.(6)), versus time for ‘full charging’. (c) Q and Hef f versus applied voltage for a given charging time. The final voltage in (b) is set to 1.4 V to ensure the maximal final charge density for both pores. This full charging is faster for the wider pore, in terms of both effective charging depth and charge accumulation. For lower voltages the charging of narrow pores can be significantly faster. The model parameters are the same as in Fig. 2.
the form of the charge density profiles at later times, as demonstrated in Fig. 2. Note that smoothening of this sharp interface results in an unphysical ‘jump’ in Q(t) at initial times (c.f. Fig. 3(b)). We now compare charging of two pores of different width (but still accomodating only one layer of ions). For chosen model parameters, the wider pore in equilibrium can accumulate more charge when fully charged (see Fig. 3(a)); its capacitance is lower, which is in line with the anomalous capacitance increase for decreasing pore width2 . The full charging of the wider pores is faster, meaning that there is more charge accumulated by the wider pore during the same charging times (see
One might be tempted to take the same route to study initially filled pores as we took for empty pores, i.e., assume the equilibrium ion densities at x = 0 that correspond to some non-zero voltage (we recall that we always (eq) assume the equilibrium ion densities ρ± (V = 0) inside the pore at t = 0). However, as discussed above, this implies that charging (and now also refilling) proceed in a front-like fashion. In case of initially filled pores, this assumption turns out to be inconsistent with the density profiles obtained at later times (c.f. Fig. 4). We thus proceed in a different way. We assume certain, not necessary equilibrium, ion densities established at the pore entrance. These ion densities change along the pore and might in fact be closer in values to the densities of the ionic double layer formed due to the flat part of the electrode, rather than to the equilibrium ion densities inside the pore. As the precise values are not known to us, we use fairly arbitrary values, provided that the charge density at x = 0 is non-zero. We emphasize that the densities obtained in the limit t → ∞ are not the equilibrium densities. Thus we focus here mainly on the initial step of the second charging stage and, realizing that the subsequent dynamics can in principle be different, we try to answer the following questions: (i) how the density profiles change during the charging process, and (ii) how the pore width influences the dynamics of charging. Our calculations show that a small ‘congestion zone’ may form at the pore entrance (see inset in Fig. 4). This congestion zone seems to spread out and move along the pore. It shall relax in the final (third) stage of the charging process, when the equilibrium densities are being established inside the pore (we do not study this process here). More importantly, Fig. 4 demonstrates that the charge density, c = ρ+ −ρ− , develops in a diffusive manner. Such diffusive-like behaviour shall characterize the entire second stage of pore charging, provided that the difference between the total ion density, ρ = ρ+ + ρ− , at the pore entrance and inside the pore at zero and non-zero applied potential is small, as compared to the corresponding difference in the charge density32 . Indeed, one may then
ACS Paragon Plus Environment
Page 5 of 9
The Journal of Physical Chemistry
(a) ρ = const V =0 12 V = 1.3 V
0.4
0.95 0.9
0.2
pore depth
9
t/τ = 2 t/τ = 1 0
6 0.7
10 20 pore depth, nm
neglect the gradient of the total ion density to obtain the diffusion equation for c(x, t), ∂t c = ∂x Def f (ρ)∂x c with the effective diffusion coefficient Def f /D = 1 + 4ρ LB Rc
∞ X sin2 (πn/2) K1 (πnRc /L) , n n=1 (7)
where Rc is the cut-off (see Eq. (3)), K1 (x) is the modified Bessel function of the second kind of first order, and we have used the local-density approximation. The effective diffusion coefficient Def f depends on the pore width L and voltage V via the total ion density ρ. It is interesting to note that Def f can be a non-monotonic function of L (see Fig. 5(a)), which is because the equilibrium total ion density, ρ, decreases with increasing L. This may suggest a faster charging by narrower pores at low voltages (c.f. Fig. 3(c)). At constant total ion density, however, the effective diffusion coefficient is a monotonic, decreasing function of L, meaning that wide pores charge faster, in terms of both accumulated charge and effective charging depth Hef f (see Fig. 5(b)). Thin lines in Fig. 5(b) show Q(t) resulting from the diffusion equation; the discrepancy with the numerical results is due to the fact that the total ion density does vary along the pore (compare Fig. 4(a)). Stage III of pore charging
It is clear that the final charging stage (stage III, see Fig. 1) does not occur for an infinitely long pore, considered in sections III A and III B. On the other hand, within the present approach we cannot study closed pores, as our interaction potential (see Eq. (4)) is not valid in the vicinity of a pore closing. Thus, to study the final stage, we consider a pore open from both sides 33 . In
charge, nC/m
0
FIG. 4. Diffusive-like charging of initially filled pores. There is a small ‘congestion’ zone close to the pore entrance (see inset) that moves along the pore and shall relax in the end of the charging process (not studied here). The Bjerrum length LB = 20 nm and ‘resolvation’ energy δE± = 15kB T . The densities inside the pore at t = 0 are set to the equilibrium densities corresponding to zero voltage. The densities at x = 0 are ρ+ /ρmax = 0.2625 and ρ− /ρmax = 0.7. The ion diameter is d = 0.7 nm and the pore width L = 0.8 nm.
C.
Def f /D
ρ/ρmax
15
0.8 0.9 pore width, nm 6
(b)
0.4
4 0.2 0
L = 0.9 nm L = 0.8 nm 0
1
2 time, t/τ
2
3
4
Hef f , nm
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
charge density, c/ρmax
5
0
FIG. 5. Charging of initially filled pores. (a) Effective diffusion coefficient Def f (see Eq. (7)) versus pore width for a constant density ρ/ρmax = 0.9625 (used at the pore entrance to obtain Q(t) in panel (b)) and for two values of the applied voltage V . (b) Accumulated charge Q and effective charging depth Hef f for two pore width. Charging of the wider pore is faster in terms of both Q and Hef f . Thin lines in (b) show Q(t) resulting from the solution of the diffusion equation with the diffusion coefficient Def f calculated from Eq. (7). The right axis in (b) shows the effective charging depth Hef f as defined by Eq. (6). The model parameters are the same as in Fig. 4.
this case, ∂x ρ± vanishes in the middle of an open pore for symmetry reasons; as this is in fact the boundary condition one would use at the pore closing, one can expect stage III for open and closed pores to be qualitatively similar. In the following we shall use the same boundary conditions on the two pore ‘entrances’, i.e., we set ρ(x = (f inal) at all times; consequently, 0, t) = ρ(x = H, t) = ρ± (eq) the initial conditions are ρ± (x, t = 0) = ρ± (V = 0) for x ∈ (0, H). Stage III turns out to be significantly faster for pores that initially (at t = 0) are empty (compare Fig. 6(a) and (b)). This is due to diffusive character of charging of filled pores which causes stage III to begin earlier as compared to empty pores (thin vertical lines in Fig. 6). More importantly, for filled pores, the charge density is being brought to its final value along the whole length of a pore only during the final stage. Evidently, the duration of stage III depends on the pore length. This can be contrasted with charging of empty pores, where the wave fronts from the two opposing pore entrances meet at a later time, and then the charge density mainly in the middle of the pore is being equilibrated during stage III. For this reason, the charging of empty pores finishes ‘abruptly’ (kinks in Fig. 6(b)). This is again in contrast to charging of filled pores where the accumulated charge
ACS Paragon Plus Environment
The Journal of Physical Chemistry
Page 6 of 9
4
(a)
II
III
25
L = 0.9 nm L = 0.8 nm 0
10 (b)
50 II
100
150
0
III empty pore Hef f , nm
charge, nC/m
filled pore
2
0
A.
50 Hef f , nm
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
charge, nC/m
6
5
0
0
50 25 0
time
10 20 time, t/τ
30
FIG. 6. Charging of pores open from both ends. The pore is initially filled in (a) and empty in (b). Thin vertical lines indicate the time when the ‘fronts’ from the two ends ‘hit’ each other, which we identify as a beginning of the final charging stage (stage III). Interestingly, stage III appears to be a few times faster for empty pores. For filled pores the ‘final’ state is not equilibrium and additional time is required for equilibration (not studied here). The total accumulated charge is different for empty and filled pores which is due to the difference in the boundary conditions, i.e. cf illed (x = 0) 6= cempty (x = 0). The right axis in (a) and the inset in (b) show the effective charging depth Hef f as defined by Eq. (6). The total pore length is H = 50 nm. The model parameters in (a) and (b) are the same as in Figs. 4 and 2, respectively.
approaches its final value in a smooth manner (Fig. 6(a)). Figure 6(a) suggests that from the viewpoint of supercapacitor’s optimization, it might be beneficial for filled pore to have a slight undercharging in favour of reducing charging times. Finally, it is important to remember that the ion densities for filled pores are not the equilibrium densities and hence an additional time is required for equilibration; we do not study this process here.
IV.
DISCUSSION OF OPTIMIZATION OF POROUS SUPERCAPACITORS
Now that we have gained some knowledge about charging dynamics, it is suggestive to discuss how porous electrodes, or more generally supercapacitors can be optimized to improve their performance, that is, to increase their capacitance, stored energy density, and, more specifically, power density.
Empty or filled pores?
The equilibrium ion densities inside pores, i.e. their wettability by ionic liquids, can be controlled by the ‘bulk’ field h(V ) (see Eq. (2)). At constant pore width, temperature, and voltage, it amounts to changing the resolvation energy and chemical potentials, which experimentally can be controlled by varying the ion and solvent concentrations in the bulk of a supercapacitor, by using multiple ion mixtures34 , or by addition of surfactants35 . It is then natural to ask whether wettable or non-wettable pores are preferred for the minimization of supercapacitor’s charging time. A na¨ıve comparison of Figs. 3 and 5, and Fig. 6 may suggest that conditions at which pores are empty at zero voltage (i.e., non-wettable pores) are better for faster charging. This conclusion may be misleading, however. Firstly, there is an unphysical jump in accumulated charge, Q, for empty pores (Fig. 3) that overestimates Q(t) at initial times, as discussed in section III A. More importantly, it appears that empty and filled pores exhibit different asymptotic behaviour in the limit of large times, essential for long/deep pores; recall that in practical applications pores can be micrometers long. We fit Q(t) shown in Figs. 3 and 5 by a simple powerlaw function, Q(t) ∼ tα , to obtain αf illed ≈ 0.54 ± 0.001 and αempty ≈ 0.44 ± 0.002 for filled and empty pores, respectively, and for pore width L = 0.8 nm (the exponents do not seem to be much affected by the pore width). Note that the value of αf illed is in a fairly good agreement with the diffusive character of charging of filled pores that leads to αdif f = 1/2. Ignoring now stage III of pore charging for simplicity, we find that the amount of charge accumulated by wettable and non-wettable pores becomes comparable when the pore depth is36 Hthr ≈ 10 ÷ 20 µm. This estimate has been done for model parameters corresponding to Figs. 3 and 5, where the charge density at the pore ‘entrance’ is lower for filled (wettable) pores: cf illed (x = 0, t)/ρmax ≈ 0.44, while cempty (x = 0, t)/ρmax ≈ 1 (full charging); one can expect lower Hthr for comparable values of c(x = 0, t). This simple analysis suggests that pores wettable by ionic liquids, and thus filled by ion pairs at zero voltage may be beneficial for fast charging of long/deep pores, while non-wettable pores may be better if pores are short.
B.
Two-step complementary optimization
It is interesting to discuss optimization of the power density (or charging times) in relation to the stored energy density. There is an opinion in the community that to optimize both quantities a combination of wide and narrow pores is required, to gain best power density from wide pores and best energy density from narrow pores. We ourselves wrote6 that “the requirements on the electrode structure to maximize each of the two characteris-
ACS Paragon Plus Environment
Page 7 of 9
The Journal of Physical Chemistry 7
−
L = Lopt Hopt charging time
FIG. 7. Two-step complementary optimization of nanoporous electrodes. In step (a), the pore width, Lopt , is chosen to maximize the stored energy density; this is done for a given ionic liquid and a specified working voltage (dash line), as described in Ref. 6. In step (b), the pore depth or length, Hopt , is chosen to satisfy the requirements on charging times (dash line).
Schematic drawing of a complementary two-step optimization is shown in Fig. 7. First, the electrode structure is optimized with respect to the stored energy density. This can be done along the lines proposed in Ref. 6, that is, for a given ionic liquid and working voltage, the stored energy density is calculated as a function of pore width, by means of, e.g., Monte Carlo or Molecular Dynamics simulations. As demonstrated in Ref. 6, the energy density is a non-monotonic function of pore width and shows a maximum at some Lopt which we identify as an optimal pore width. We do not expect the pore depth to have any significant influence on Lopt as long as pores are sufficiently long. The second step seems ‘trivial’. For a given optimal pore width, the optimal pore depth/length is chosen to obtain the required charging time. From computational point of view, this can be done by performing Molecular Dynamics simulations for short pores and then ‘extrapolating’ the results for longer pores, similarly as has been discussed in section IV A (simulations of micrometer long pores do not seem realistic at the moment). In this respect also the development of novel methods capable of producing quantitatively reliable results for charging dynamics of micrometer-long pores is important.
channels
channels
electrode
channels
channels
electrode
−
FIG. 8. Multi-layered configuration aimed to increase the capacitance, store energy and power density of a supercapacitor. (a) The positive electrodes provide dead-end pores of slit or other geometry, whereas the negative electrode is a construction providing channels open from both sides. The electrodes are separated by non-conducting, ion-permeable membranes to prevent short cuts. The big rectangle embraces the whole construction and is filled with an electrolyte. (b) More porous channels can be added to form a multi-layered supercapacitors built of interdigitated ‘conventional’ porous electrodes (labelled as “electrodes”) and electrodes with pores open from both sides (labelled as “channels”); separating membranes are not shown for brevity.
C.
tics could be competing”. Here we propose a simpler and more natural point of view in which competing is replaced by complementary.
electrode
membrane
Hopt
channels
− +
working voltage (b)
electrode
porous channels
Lopt
Lopt
(b)
+
+
(a) porous electrode
optimal pore width
(a)
pore depth/length
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Multilayered supercapacitor with porous nano-channels
Motivated by our study of pores open from both sides (see section III C), it is instructive to suggest the use of ‘porous nano-channels’ in a supercapacitor to improve its performance. They can be put in between the (porous) electrodes, as shown schematically in Fig. 8. As compared to the conventional supercapacitor, this way one increases the stored energy and capacitance, as electrodes form parallel capacitors, and, additionally, one gains a decrease in charging times (approximately twice for the same energy density). Note that for best performance, the charging time and capacitance of electrodes and channels must be adjusted to be the same. The three layer system shown in Fig. 8(a) can be extended by adding more porous channels and arranging them sequentially, as depicted in Fig. 8(b). Such ‘multilayered’ supercapacitors are essentially equivalent to a set of conventional supercapacitors connected in parallel. Their advantage is, however, that they can be made more compact, thus reducing the total mass and volume at the same other characteristics.
V.
CONCLUDING REMARKS
We have considered a simple mean-field-like model to study charging dynamics of nano-sized porous electrodes. The dynamics are defined by the continuity equation (Eq. (5)) and by appropriate boundary and initial conditions (section II B). Pores were classified and studied separately based on whether they were ‘initially’ empty
ACS Paragon Plus Environment
The Journal of Physical Chemistry
Page 8 of 9 8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
or filled. The initial (at time t = 0) state is related to the pore wettability: pores wettable by ionic liquids are filled with ion pairs (at zero voltage), while non-wettable pores are ‘empty’28 . Our findings can be summarized as: - With some possible exceptions (see Fig. 3(c)), it seems to be generally true (see discussion below) that wider pores charge faster, in the sense of accumulating charge, regardless how well they are wetted by ionic liquids (Figs. 3(b) and 5(b)). - Charging of empty (i.e. non-wettable) pores proceeds in a front-like manner (Fig. 2). - Charging of filled (i.e. wettable) pores seems to be generally diffusive (see Eq. (7) and Fig. 4(b)). - Charging of wettable pores appears to be initially slower but asymptotically, i.e. in the limit of large charging times, faster than charging of nonwettable pores (see section IV A). - Finalization of the charging process (i.e. stage III, see Fig. 1) seems to be much faster for empty pores (Fig. 6). We have also discussed two possible improvements to enhance the performance of nanoporous supercapacitors, including - The use of ‘porous nano-channels’ in a ‘multilayered’ supercapacitor aimed to decrease its volume/size, and increase capacitance and power density (section IV C and Fig. 8). - Two-step complementary optimization in which (a) the pore width is chosen to maximize the energy density, and (b) the pore depth/length is chosen to obtain the required charging time (section IV B and Fig. 7). A few comments are in order. The main drawback of this study is that our two dimensional mobility parameter Γ (and hence the corresponding diffusion constant D = kB T Γ) is not known, and thus little can be said about real charging times. Additionally, the diffusion coefficient, as well as the dielectric constant inside the pores18 , may vary as a function of pore width. Not much is known about the pore width dependence of Γ, one can speculate, however, that Γ decreases with increasing pore width, at least for ballistic transport; this can modify our conclusion that wider pores charge faster. It would thus be extremely interesting, and beneficial from the viewpoint of supercapacitor’s optimization, to study this point in more details. There are reasons to believe that charging of sufficiently wide pores proceed in a diffusive way, independently of pore wettability37 . Therefore it would be very
interesting to look at intermediate-width pores and a possible transition between the diffusive and front-like behaviour. Our model, however, consists of a single ionic liquid layer, hence it is applicable only to very narrow pores, comparable in size with the ion diameter (the ‘three-dimensionality’ is accounted for in an approximate way via the expression for entropy4 ). It seems thus highly desirable to extend this study to the case of wider pores. This can be done by either expanding the existing model to three-dimensions, or by formulating a more robust time-dependent density functional theory22–27 . Molecular Dynamics simulations may as well be an option; such studies would require significant computational resources, however. Another critical issue is the density dependence of the mobility parameter Γ. In our comparison of charging of filled and empty pores (see section IV A) we have silently assumed that Γ is the same in both cases. Although this does not have to be true, and most probably is not, it shall not affect our conclusions in any qualitative way. Indeed, it is not difficult to see that a different Γ shall merely influence the magnitude Q0 in the asymptotic behaviour, Q(t) = Q0 tα , while the exponent α shall remain the same. Our final remark concerns the second ‘complementary’ optimization step (see section IV B), which consists of computing charging times versus pore depth for a given porous material. This is easy to say but not so easy to accomplish. Indeed, we mentioned in section IV B that Molecular Dynamics (MD) simulations do not seem to be appropriate for this purpose, in view of very large depth of realistic pores. We suggested instead a simple ‘extrapolation’ of the results for short pores. This might be quantitatively unreliable, however. A more reliable approach could be the combination of MD simulations and dynamical density functional theory (DDFT). For instance, one can employ MD simulations to calculate the pore-width and density dependent diffusion coefficient (or more generally diffusion matrix), which can then be used in much faster yet sufficiently robust DDFT to compute charging times for desirable pore lengths/depths. Such an approach seems also to have a potential to be developed to studies of more realistic porous materials. ACKNOWLEDGMENTS
We thank Gleb Oshanin and Alpha Lee for impetuous debates, and Rui Quiao for interesting exchange of views. S. K. is grateful to Markus Rauscher, Stefan Kesselheim, Christian Holm, and Markus Bier for fruitful discussions. We also thank Carlos P´erez, Majid Beidaghi and Yury Gogotsi for critical reading of the manuscript and for useful comments. This work was supported by Grant EP/H004319/1 from the Engineering and Physical Science Research Council.
ACS Paragon Plus Environment
Page 9 of 9
The Journal of Physical Chemistry 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1
2
3
4 5 6
7
8 9
10 11 12 13
14
15
16
17
18
19
20
B. E. Conway, Electrochemical Capacitors: Scientific Fundamentals and Technological Applications (Kluwer, 1999). C. Largeot, C. Portet, J. Chmiola, P.-L. Taberna, Y. Gogotsi, and P. Simon, J. Am. Chem. Soc. 130, 2730 (2008). R. Mysyk, E. Raymundo-Pi˜ nero, and F. B´eguin, Electrochem. Comm. 11, 554 (2009). S. Kondrat and A. Kornyshev, J. Phys.: Condens. Matter 23, 022201 (2011); 25, 119501 (2013). S. Kondrat, N. Georgi, M. V. Fedorov, and A. A. Kornyshev, Phys. Chem. Chem. Phys. 13, 11359 (2011). S. Kondrat, C. R. P´erez, V. Presser, Y. Gogotsi, and A. A. Kornyshev, Energy Environ. Sci. 5, 6474 (2012). C. Merlet, B. Rotenberg, P. A. Madden, P.-L. Taberna, P. Simon, Y. Gogotsi, and M. Salanne, Nature Materials 11, 306 (2012). H. Sakaguchi and R. Baba, Phys. Rev. E 76, 011501 (2007). P. M. Biesheuvel and M. Z. Bazant, Phys. Rev. E 81, 031502 (2010). N. N. Rajput, J. Monk, and F. R. Hung, J. Phys. Chem. C 116, 14504 (2012). R. de Levie, Electrochim. Acta 8, 751 (1963). M. Z. Bazant, K. Thornton, and A. Ajdari, Phys. Rev. E 70, 021506 (2004). M. S. Kilic, M. Z. Bazant, and A. Ajdari, Phys. Rev. E 75, 021502 (2007); 75, 021503 (2007). Strictly speaking one can associate the pore depth with the electrode thickness only for templated carbons, as porous carbons consist mainly of shorter, inter-connected pores. Nevertheless, in the first approximation one can neglect the pore branchings/connections, and consider the charging of a single but longer pore. K. Kiyohara and K. Asaka, J. Phys. Chem. C 111, 15903 (2007); K. Kiyohara, T. Sugino, and K. Asaka, J. Chem. Phys. 132, 144705 (2010); 134, 154710 (2011). B. Skinner, T. Chen, M. S. Loth, and B. I. Shklovskii, Phys. Rev. E 83, 056102 (2011); B. Skinner, M. M. Fogler, and B. I. Shklovskii, Phys. Rev. B 84, 235133 (2011). As noted in Ref. 4, the value of the Bjerrum length in the bulk is of the order of 5 nm for a typical ionic liquid and 0.7 nm for aqueous electrolyte solutions at room temperature. In nanopores the dielectric constant is reduced, and is determined by an effective polarizability of a quasi twodimensional layer of ions. Its value is not known, but one can expect it to be in the range between 2 and 5, which gives the Bjerrum length between 11 and 28 nm at room temperature. As shown elsewhere (S. Kondrat, A. Kornyshev, F. Stoeckli and T. A. Centeno, to be published), the pore-width dependent dielectric constant may lead to a constant capacitance in a wide range of pore widths, instead of its anomalous increase below 1 nm. See for instance W. R. Smythe, Static and Dynamic Electricity (McGraw Hill, 1939). In can be shown by means of DFT calculations (Markus Bier, unpublished) that this approximation also leads to incorrect correlation functions for r < bαβ .
21
22 23 24
25 26 27
28
29
30
31
32
33
34
35 36
37
I. Borukhov, D. Andelman, and H. Orland, Phys. Rev. Lett. 79, 435 (1997). W. Dieterich, H. L. Frisch, and A. Majhofer, Z. Phys. B 78, 317 (1990). D. S. Dean, J. Phys. A 29, L613 (1996). U. M. B. Marconi and P. Tarazona, J. Chem. Phys. 110, 8032 (1999). A. J. Archer and R. Evans, J. Chem. Phys. 121, 4246 (2004). A. J. Archer and M. Rauscher, J. Phys. A: Math. Gen. 37, 9325 (2004). P. Espa˜ nol and H. L¨ owen, J. Chem. Phys. 131, 244101 (2009). By ‘empty’ we do not mean vacuum-empty pores but rather that pores are mainly filled with a solute. The presence of solute seems to be one reason why the ions may not be willing to enter the pores (at zero voltage). GNU Scientific Library, http://www.gnu.org/software/gs l/. We have checked this for the cut-off Rc & 1.2. There are some numerical difficulties for smaller values of Rc related to the evaluation of integrable singularities. We use Rc = 0.6 in this work as it reflects most closely in values the equilibrium results of the density-dependent cut-off used in Ref. 4. Other quantities can as well be chosen to describe the pore charging dynamics. For example, some people choose to show the distance at which the charge density is 1/2 of the ‘desired’ charge density, i.e. c(x = 0)/2 in our notations. We find the effective pore depth Hef f (t) a more natural candidate to discuss the charging process, especially in the context of charging of empty pores. This shall be true for room temperature ionic liquids characterized by the so-called ‘swapping charging type’ as discussed in Refs. 4 and 5 and confirmed by more elaborate Molecular Dynamics simulations of Ref. 7. We choose x = 0 (and x = H) sufficiently far from the real pore entrance, see Fig. 1, so that its effect on the interaction potential, Eq. (4), can be safely neglected; evidently, this cannot be done for pore closing. R. Lin, P.-L. Taberna, S. Fantini, V. Presser, C. R. P´erez, F. Malbosc, N. L. Rupesinghe, K. B. K. Teo, Y. Gogotsi, and P. Simon, J. Phys. Chem. Lett. 2, 2396 (2011). K. Fic, G. Lot, and E. Frackowiak, Electrochim. Acta 55, 7484 (2010); 60, 206 (2011). To estimate the threshold pore depth, Hthr , we first calculate the threshold charging time, tthr , by equating Q(t) = Q0 tα for filled and empty pores and using the corresponding values of Q0 and α. Then tthr is used to evaluate Hef f from Eq. (6). This way we obtain two different values of Hef f that define us the range of pore depths within which the accumulated charge of initially empty and filled pores becomes comparable. The wettability dependence of charging of wide pores has not been studied. One can speculate however that wide pores can never be empty in virtue of the bulk-like region formed in the middled of the pores. Thus their charging shall be well described by the transmission-line-type model that leads to a diffusive charging8 .
ACS Paragon Plus Environment
