How Do Pseudocapacitors Store Energy? Theoretical Analysis and

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How Do Pseudocapacitors Store Energy? Theoretical Analysis and Experimental Illustration Cyrille Costentin,* Thomas R. Porter, and Jean-Michel Savéant* Laboratoire d’Electrochimie Moléculaire, Unité Mixte de Recherche Université CNRS 7591, Université Paris Diderot, Sorbonne Paris Cité, Bâtiment Lavoisier, 15 rue Jean de Baïf, 75205 Paris Cedex 13, France S Supporting Information *

ABSTRACT: Batteries and electrochemical double layer charging capacitors are two classical means of storing electrical energy. These two types of charge storage can be unambiguously distinguished from one another by the shape and scan-rate dependence of their cyclic voltammetric (CV) current−potential responses. The former shows peak-shaped current−potential responses, proportional to the scan rate v or to v1/2, whereas the latter displays a quasi-rectangular response proportional to the scan rate. On the contrary, the notion of pseudocapacitance, popularized in the 1980s and 1990s for metal oxide systems, has been used to describe a charge storage process that is faradaic in nature yet displays capacitive CV signatures. It has been speculated that a quasi-rectangular CV response resembling that of a truly capacitive response arises from a series of faradaic redox couples with a distribution of potentials, yet this idea has never been justified theoretically. We address this problem by first showing theoretically that this distribution-of-potentials approach is closely equivalent to the more physically meaningful consideration of concentrationdependent activity coefficients resulting from interactions between reactants. The result of the ensuing analysis is that, in either case, the CV responses never yield a quasi-rectangular response ∝ ν, identical to that of double layer charging. Instead, broadened peak-shaped responses are obtained. It follows that whenever a quasi-rectangular CV response proportional to scan rate is observed, such reputed pseudocapacitive behaviors should in fact be ascribed to truly capacitive double layer charging. We compare these results qualitatively with pseudocapacitor reports taken from the literature, including the classic RuO2 and MnO2 examples, and we present a quantitative analysis with phosphate cobalt oxide films. Our conclusions do not invalidate the numerous experimental studies carried out under the pseudocapacitance banner but rather provide a correct framework for their interpretation, allowing the dissection and optimization of charging rates on sound bases. KEYWORDS: electrochemical capacitors, pseudocapacitance, double-layer charging, faradaic reactions, cyclic voltammetry

1. INTRODUCTION There are two electrochemical ways to store electrical energy: batteries and electrochemical capacitors.1 In the first case, faradaic processes are involved: that is, electron transfer occurs across the electrode surface to or from reactants present in solution or adsorbed at the electrode surface. This includes systems where the redox sites are embedded in an electronically conductive material and can involve counterion insertion/ desertion during the charging process. In the second category, no faradaic process occurs; that is, no electron transfer across the surface takes place. Charge storage in this case is the result of charging the electrochemical double layer at the interface between the electrode and the bathing solution. If the surface area of this interface has been increased substantially by recourse to electrode materials with large specific surface area, as is typically the case with various form of carbons,2−6 then ultracapacitors or supercapacitors are obtained. While double layer capacitors typically store less charge than batteries, they can be charged and discharged at very fast rates. The cyclic voltammetric (CV) signatures of faradaic charging processes are quite different from those associated with double layer charging processes, as sketched in Figure 1. Other © 2017 American Chemical Society

transient techniques, such as potentiostatic chronamperometry or galvanostatic chronopotentiometry,7 could be used as well, but CV is diagnostically more convenient.8 Double layer charging and discharging9 give rise to the classical capacitance charging CV current−potential responses8a as represented in Figure 1a. In this case, the double layer capacitance is approximately constant within the potential excursion range. The charging curve shows a quasi-rectangular shape, and the whole curve is proportional to the scan rate. Surface faradaic CV current−potential responses, are quite different: they show peaks as represented in Figure 1b.8b,10,11 The peak current, like the capacitive plateau current, is proportional to the scan rate, but the difference in shape makes them readily distinguishable. Solution faradaic CV current−potential responses also display peak shapes (Figure 1c), but the current now depends on reactant diffusion to and from the electrode surface.12−14 Peak currents are then proportional to the square root of the scan rate. Received: November 3, 2016 Accepted: February 14, 2017 Published: February 14, 2017 8649

DOI: 10.1021/acsami.6b14100 ACS Appl. Mater. Interfaces 2017, 9, 8649−8658

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ACS Applied Materials & Interfaces

pseudocapacitance: “There are an increasing number of studies regarding active electrode materials that undergo faradaic reactions but are used for electrochemical capacitor applications. Unfortunately, some of these materials are described as ‘pseudocapacitive’ materials despite the fact that their electrochemical signature (e.g., cyclic voltammogram and charge/ discharge curve) is analogous to that of a ‘battery’ material, as commonly observed for Ni(OH)2 and cobalt oxides in KOH electrolyte. Conversely, true pseudocapacitive electrode materials such as MnO2 display electrochemical behavior typical of that observed for a capacitive carbon electrode.” [italics ours]. That is to say, true pseudocapacitive materials display a quasirectangular CV response proportional to scan rate. A clear example of a MnO2 electrode displaying this typical CV response can be seen,19 whereas a much broader definition encompassing underpotential deposition, redox pseudocapacitance, and intercalation pseudocapacitance is considered in ref 20. The common CV feature in these three cases is that the current is proportional to scan rate and that “the shape is rectangular and if peaks are present, they are broad and exhibit a small peak-to-peak voltage separation”.20 Such a broad definition covers very diverse situations that all occur “due to the relationship between potential and the extent of charge that develops as a result of adsorption/desorption processes at the electrode/electrolyte interface or within the inner surface of a material where E is the potential and X is the extent of fractional coverage of the surface or inner structure.”20: E ∼ E° +

RT ⎛⎜ 1 − X ⎞⎟ ln F ⎝ X ⎠

This equation is a Nernst-like relationship, which emphasizes the faradaic character of the so-defined pseudocapacitance. Whether we consider the strict definition of pseudocapacitance from ref 18 or the broader one from ref 20, they both include cases where the CV response is quasi-rectangular, as with capacitive double layer charging carbon electrodes. In spite of the very large number of studies invoking the notion of pseudocapacitance, notably involving those devoted to transition metal oxides (see ref 18 and all references in the corresponding J. Electrochem. Soc. 2015, 162 Special issue on electrochemical capacitors: fundamentals to applications), the reasons that faradaic reactions can display quasi-rectangular capacitive electrochemical signatures remain unclear.21. A first explanatory indication is contained in the following quote from ref 16: “it is believed that the capacitive behavior arises on account of several overlapping redox processes involving proton and electron injection (or removal) that remain highly reversible because no phase changes arise”. The same idea is taken up with no further analysis (the same idea is taken up with no further analysis in Figure 6 of ref 22), as well as in the modeling of MnO2 pseudocapacitance based on an ad hoc linear dependence between the equilibrium potential and the oxidation state.23 This analysis is in puzzling contrast with the Nernst-like relationship, which is supposedly at the root of all pseudocapacitive behaviors as mentioned above. This issue of whether it is really possible to obtain a quasirectangular capacitive behavior from the superposition of several overlapping surface faradaic processes is the first we address. At this occasion we will show that the effect of superposing several overlapping redox couples is closely equivalent to the introduction of concentration-dependent activity coefficients resulting from interactions between the redox reactant molecules. The latter approach provides a more

Figure 1. Cyclic voltammetric current potential responses: i, current; E, electrode potential; E°, standard potential of the surface or solution redox couple; v, scan rate; S, electrode surface area; F, Faraday constant; T, absolute temperature. (a) Double layer charging current: Cd, double layer differential capacitance; E°, middle of the potential excursion. (b) Faradaic response of a surface redox couple: Γm, surface concentration; E°, standard potential of surface redox couple. (c) Faradaic response of solution redox couple: C°, solution concentration; E°, standard potential of surface redox couple; D, reactant diffusion coefficient.

Distinction between faradaic and double layer charging has been done so far by means of their CV signatures (wave shape and scan rate dependence of plateau or peak currents) and is presently perfectly unambiguous. This is not the case with pseudocapacitors and with the notion of pseudocapacitance, whose definitions are the object of puzzlingly divergent views. This notion was originally defined as a faradaic process (as the ones in Figure 1b,c) that has, at the same time, a capacitive CV signature; that is, it shows a quasi-rectangular CV response proportional to scan rate, similar to that of a double layer charging response (as in Figure 1a).15−17 Reference 18 insists on the strict adherence to this original definition of 8650


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type shown in Figure 1b is the CV signature of a faradaic reaction involving a fast redox couple P/Q, where the oxidized form P and the reduced form Q are both adsorbed on the electrode surface and behave in an ideal manner, thus obeying the Nernst law:

meaningful physical picture than the putative superposition of several overlapping redox processes. The former has more the flavor of an ad hoc explanation in the absence of physical justification for the existence of a standard potential distribution, the number of standard potentials it contains, the potential gap between them, and the form of the distribution law. As a first experimental illustration, we qualitatively examine, in section 2.2, two emblematic metal oxide systems, RuO2 and MnO2. The next question we will address, in section 2.3, is the transition from a diffusion-controlled faradaic CV response to a surface faradaic CV response, in the framework of the reaction scheme shown in Scheme 1:

E = E° +

Here ΓP and ΓQ are surface concentrations of the oxidized and reduced forms of the redox couple, respectively, and E° is the standard potential of the couple. The P/Q formulation of the reaction under examination is general in the sense that it may involve additional species, such an acid−base couple, provided equilibrium is also achieved when these additional species are taken into consideration. In such a case, their concentrations would appear in the Nernst law. However, ideality, as expressed by the applicability of the above Nernst law, may not be always achieved in most practical cases. Such a nonideal situation occurs when interactions between the reactant molecules lead to concentration-dependent activity coefficients. These activity coefficients, γP and γQ, then appear in the expression of the Nernst law:

Scheme 1: P(ads) + e− + Z+ ⇌ QZ(ads)

where the electron transfer taking place at the electronic conductive film/solution interface is accompanied by insertion of a charge-compensating cation Z+, possibly a proton or an alkaline cation. Two cases will be examined. In the first case (section 2.3.1), diffusion-like Z+-coupled electron hopping in the film accompanies the interfacial reaction (e.g., as in Li+ or Na + insertion into metal oxide electrodes) while the concentration of Z+ in the solution is kept constant. In the second case (section 2.3.2), diffusion of Z+ in the solution accompanies interfacial electron transfer, while electron and Z+ transport through the film are fast. In these two cases, particular attention will be devoted to the effect of the scan rate (v) as a diagnostic criterion of the respective role of diffusion (proportional to v1/2) and surface reaction (proportional to v). The magnitude of the scan rate also governs, together with other parameters, the transition from diffusion control to surface reaction behavior. For the sake of simplicity, these two analyses will be carried out under conditions where the interactions between reactants can be neglected and in the absence of structural modifications in the film as well as electron transfer kinetics interference. In section 2.4, a more complicated reaction scheme will then be examined, involving the reversible reaction shown in Scheme 2: Scheme 2: −

RT ⎛ ΓP ⎞ ln⎜⎜ ⎟⎟ F ⎝ ΓQ ⎠

E = E° +

⎛ RT ⎜ γP ΓP ln⎜ F ⎝ γQ ΓQ

⎞ ⎟ ⎟ ⎠

The average interaction energy for contribution to the chemical potential of surface species may be approximated by a linear function of P and Q coverages, ΓP/Γm and ΓQ/Γm, where Γm is the maximal surface concentration the P and Q molecules can occupy, implying that (ΓP/Γm) + (ΓQ/Γm) = 1.26−28 That is, after introduction of the interaction coefficients aP, aQ, and aPQ: ln(γP) = 2aP(ΓP/Γm) + 2aPQ (ΓQ /Γm)

and ln(γQ ) = 2aQ (ΓQ /Γm) + 2aPQ (ΓP/Γm)

The following expression of P and Q coverages ensues: ΓQ Γm

=1− =

−

P + ZH + e ⇌ QH + Z

ΓP Γm

1 ΓQ ⎤ ⎡ F ⎡ ⎤ 1 + exp⎣ RT (E − E°app )⎦ exp⎢− (aQ + aP − 2aPQ ) 1 − 2 Γ ⎥ ⎣ m ⎦

(

)

(1)

In this case, the reactant interactions (or equivalently standard potential distribution) are taken into account in an effort to provide the necessary tools to analyze the charge−discharge behavior of phosphate cobalt oxide films. These films are wellknown as O2 evolution water oxidation catalysts.24 As shown in a recent study,25 mostly dedicated to the mechanism of the catalytic reaction, they display two proton-coupled faradaic waves at potentials below the catalytic wave that depend on the presence and concentration of phosphate buffer in the bathing solution. It is therefore possible to turn faradaic reactions on or off by varying the concentration of phosphate buffer. This, in addition to CV scan rate, allows one to modulate the faradaic contributions to the overall current.

where E°app is an apparent standard defined as E°app = E° +

RT (aP − aQ ) F

When it is taken into account that the potential varies linearly with time (dE/dt = ±v), the current is obtained from the variation of P and Q coverage with time: d(ΓQ /Γm) i = FS Γm dt Fv =± RT

2. RESULTS AND DISCUSSION 2.1. Is It Possible To Obtain Capacitive Behavior from the Superposition of Several Overlapping Surface Faradaic Processes? The current−potential response of the

⎡ ⎢⎣1 −

( )⎤⎥⎦( ) ΓQ

ΓQ

Γm

Γm

⎡ 1 + 2(aQ + aP − 2aPQ )⎢1 − ⎣

( )⎤⎥⎦( ) ΓQ

ΓQ

Γm

Γm

(2)

where i is current and S is electrode surface area. 8651


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ACS Applied Materials & Interfaces Combination of eqs 1 and (2) and integration, taking into account the initial condition (ΓQ/Γm)t=0 = 0, allows the derivation of the current−potential CV responses as functions of the interaction parameter, aQ + aP − 2aPQ, and of the apparent standard potential, E°app, as described in Supporting Information. Positive values of this parameter lead to broadening of the surface wave as seen in Figure a, which also displays the CV responses as a function of the interaction parameter. We now show that taking into account such interactions is closely equivalent to considering the superposition of several overlapping redox couples believed, as indicated earlier, to be at the origin of pseudocapacitance behaviors. Equation 1 indicates that, due to interactions, the surface coverage ΓQ/Γm at a given potential is equivalent to the surface coverage expected for an ideal couple having a standard potential equal to E* = E°app + (RT/F)(aQ + aP − 2aPQ)(1 − 2ΓQ/Γm). Peak broadening occurs when aQ + aP − 2aPQ is positive. Under these conditions, in the E > E°app potential region, E* > E°app, and vice versa in the E < E°app potential region, E* < E°app. One is thus led to the idea that peak broadening due to interactions between reactants may be approximately described by a series of redox couples with distributed standard potentials. This series is defined as an equally distributed set of standard potentials, ranging from −ΔE° to ΔE° and centered at E°app; the probability of each is (2FΔE°/RT)−1, with FΔE°/RT = 2(aQ + aP − 2aPQ) (only considering positive values of aQ + aP − 2aPQ interaction parameter):

Figure 2. Current−potential CV responses as a function of the parameter 2(aQ + aP − 2aPQ) = FΔE°/RT = 0.1 (blue), 1 (red), 2 (green), 3 (yellow), 4 (gray), and 5 (magenta). (a) Calculated from a combination of eqs 1 and 2, corresponding to interactions between the reactant molecules leading to concentration-dependent activity coefficients. (b) Calculated from eq 4, corresponding to the square standard potential distribution.

current response under the form of an apparent capacitance (i/ v), even though it has nothing to do with a true capacitor charging. In other words, the CV current−potential responses of surfaces redox couples may be expressed as apparent capacitance−potential responses, but they are clearly distinguished from true capacitive responses (peak instead of quasirectangle). If pseudocapacitance was simply defined as an apparent capacitance, regardless of the shape of the CV signature, there would be no problem. However, as discussed earlier, this is not consistent with any of the currently used working definitions of pseudocapacitance. It is well possible that, in many cases, surface faradaic peaks may add to capacitive double layer charging, thus improving the overall charge loading. Figure 3 shows two such cases. The

F ΔE ° / RT

ΓQ Γm

=

RT 2F

∫

1 dζ F ⎡ 1 + exp⎣ RT (E − E°app )⎤⎦ exp ζ

−F ΔE ° / RT

(3)

The current−potential response may therefore be expressed as i Fv FS Γm RT

=±

⎡ F ⎤ RT /F exp⎢ (E − E°app )⎥ ⎣ ⎦ 2ΔE° RT

⎧ ⎪ ×⎨ ⎪ 1 + exp ⎩

( FRTΔE ° )

exp

( FRTΔE ° ) exp⎡⎣ RTF (E − E°app)⎤⎦ F ΔE ° ⎫ exp(− RT ) ⎪ ⎬ − F ΔE ° F 1 + exp(− RT ) exp⎡⎣ RT (E − E°app )⎤⎦ ⎪ ⎭ (4)

The CV responses computed by one method (eqs 1 and 2 and initial conditions) or the other (eq 4) give very similar results (Figure 2). The second method, albeit less rigorous, leads to a closed-form expression of the current, therefore allowing simpler calculations than the first. Whichever of these two descriptions is taken into consideration, comparison of Figure 1a with Figure 2 shows that no capacitive response can be observed with surface faradaic processes. Surface faradaic processes show only peaks, which are more or less broadened according the degree of reactant interactions, with an exponential shape at the foot of the wave very different from an RC-like capacitor charging profile. It is worth noting, however, that the proportionality of the current to the scan rate makes it possible to express the

Figure 3. Examples of current−potential CV responses mixing double layer charging and surface faradaic processes. (a) Single faradaic surface reaction with 2(aQ + aP − 2aPQ) = FΔE°/RT = 2. (b) Two successive faradaic surface reactions with 2(aQ + aP − 2aPQ) = FΔE°/ RT = 1 for each.

dashed lines suggest the way in which the two contributions can be deconvoluted. The whole current−potential trace is proportional to the scan rate, showing that shape recognition is the decisive diagnostic criterion in these cases. 2.2. Examples from the Literature: Two Emblematic Metal Oxide Systems, RuO2 and MnO2. Manganese dioxide is one of the most-investigated materials referred to as pseudocapacitive. Since the beginning,29 many cyclic voltam8652


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requires charge compensation by insertion of a cation Z+, possibly a proton or an alkaline cation, and thus takes place at the film/solution interface and corresponds to the reaction shown in Scheme 1. Oxidation of P(ads) embedded in the conductive bulk film requires Z+ transport through a diffusionlike hopping process, as described in Supporting Information. This process may be kinetically limiting. Alternatively, Z+ diffusion in solution to reach film/solution interface may also be kinetically limiting. 2.3.1. Case 1: Z+-Coupled Electron Hopping within the Film Is Kinetically Limiting. We start with the case where diffusion-like Z+-coupled electron hopping accompanies the interfacial reaction (e.g., Li+ or Na+ insertion into metal oxide electrodes), while the concentration of Z+ in the solution is kept constant. This is related to a promising series of metal oxide films with alkaline metal insertion, which currently attracts active interest.18,20 As shown in Supporting Information, the competition between the surface reaction behavior and diffusion inside the film of the inserting cation is governed by the following parameter:

mograms of diverse MnO2 materials have been reported in the literature in various conditions (supporting electrolyte, scan rate, material structure, etc.). In most cases (see for example Figure 1 in ref 30, Figure 7 in ref 19 and Figure in ref 23), the CV is quasi-rectangular, with a flat current versus potential domain over 0.8−1 V. Increasing deviation from a rectangular shape, possibly due to ohmic drop, is clearly observed as scan rate is increased ((see Figure 4 in ref 23). Depending on the structure of the material, a broad quasi-reversible redox feature can be (see Figure 3C in ref 31 corresponding to the cryptomelane MnO2), above the quasi-rectangle, similar to the CV curve shown in Figure 3a. These data qualitatively match our analysis. Indeed, if the observed flat current−potential curve, covering a 0.8 V potential range, were to be assigned to a faradaic process, this would be equivalent to a standard potential distribution, representing the reactant interactions, spreading out over a range as large as 0.8 V [2(aQ + aP − 2aPQ) = FΔE°/RT = 32], which is an obviously arbitrary and absurd figure. It follows that this flat current should rather be assigned to capacitive double layer charging. Spectroscopic measurements [X-ray photoelectron spectroscopy (XPS)30 and X-ray absorption spectroscopy (XAS)32] tracking manganese oxidation number change have been used to support the claim that the pseudocapacitive behavior of MnO2 is of faradaic nature. However, XPS measurements are ex situ, and the modification of the XPS signal cannot be quantitatively correlated to the charge stored upon polarization. XAS data are in situ measurements, but the signal acquisition corresponding to a change of Mn oxidation number corresponds to a time scale ca. 2 orders of magnitude higher than the cyclic voltammetry time scale. Moreover, the oxidation number change shows irreversibility, whereas the cyclic voltammetry response is reversible over multiple cycles. It thus follows that none of the reported spectroscopic observations proves that the MnO2 capacitive charging is due to faradaic processes. A second example is ruthenium dioxide (RuO2), which was the first material to be described as having pseudocapacitive properties.33 This material accordingly exhibits an almost rectangular CV response (see, e.g., Figures 1 in refs 33 or 34) and shows metallic conduction associated with capacitive double layer charging electrochemical behavior.35 These observations also fall in line with recent theoretical calculations supporting a description of conduction as involving an electronic band structure rather than localized redox states.36 Reversible redox features are, however, also observed on top of the capacitive CV response (see Figure 2 in ref 35), in agreement with our above theoretical analysis. It is also remarkable that these faradaic contributions increasingly appear upon lowering scan rate and upon addition of increasing amounts of water to layered ruthenic acid hydrate, which is consistent with the role of solution diffusion in surface faradaic coupled processes (vide infra). At this stage, we may conclude that the main features of CV responses reported for the two classical MnO2 and RuO2 systems are in good agreement with the predictions of our theoretical analysis. A more detailed and more accurate test, described in section 2.4, will involve phosphate cobalt oxide films, after the respective role of diffusion and surface reactions has been clarified in section 2.3. 2.3. Diffusion and Surface Reactions. We consider an electronic conductive film deposited on an underlying electrode. A homogeneous distribution of immobile redox centers P(ads) is assumed within the bulk film. Oxidation of P(ads)

where df is the film thickness and Df is the diffusion coefficient or equivalent diffusion coefficient (see Supporting Information) of the inserting cation in the film. A diffusion-controlled CV response is obtained for thick films and fast scan rates, and a surface wave is obtained for thin films and slow scan rates. The transition between the two limiting behaviors is shown as a succession of dimensionless CV responses (Figure 4a; see Supporting Information for calculation details). This is also shown as a variation of the peak current, which changes as a dif,film dif,film function of psurf (Figure 4a′), where psurf contains contributions from film thickness, diffusion coefficient, and scan rate. CbZ is the concentration of Z+ in bulk solution; C° is the standard reference concentration, taken as equal to 1 M; and Cm is the total concentration of redox species in the film (Γm = Cmdf). All symbols are additionally gathered in a Glossary of Symbols at the beginning of Supporting Information. The red curves in Figure 4a′ apply to the case where the double layer charging current is negligible compared to the faradaic current. The dotted line represents the limiting case where control by the surface reaction prevails, whereas the dashed line represents the converse limit where control is by diffusion. The blue curves in Figure 4a′ are correspond to the case where the double layer charging current is no longer negligible relative to the faradaic current and the film thickness remains constant (meaning that scan rate is the only varying parameter in pdif,film surf ). The global CV responses are then of the type shown in Figure 3, where the faradaic waves develop on top of quasihorizontal capacitive current. We have represented the variations of peak current for an arbitrary value of the RTCd/ F2SCmdf parameter (0.1) that measures the relative magnitudes of the double layer charging current and the faradaic current. Cd is the differential capacitance of the double layer. Other values of this parameter would simply results in vertical translation of the solid blue curve. Using these curves, and still in the framework of constant film thickness, we may now examine the validity of eq 5, introduced to estimate the occurrence of pseudocapacitance versus ordinary faradaic currents:37 8653


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A diffusion-controlled CV response is obtained with low Z+ concentration in solution and fast CV scans, and a surface wave is obtained with high Z+ concentrations in solution and slow CV scans. The transition between the two limiting behaviors is shown as a succession of dimensionless CV responses (Figure 4b). This is also shown as the variation of peak current (Figure 4b′) with the parameter pdif,sol surf and, through it, with the two operational parameters, Z+ solution concentration and scan rate. The red curves in Figure 4b′ apply to the case where the double layer charging current is negligible compared to the faradaic current. The dotted line represents the limiting case where control by the surface reaction prevails, while the dashed line represents the opposite limit where control is by diffusion. The blue curves in Figure 4b′ illustrate the case where the double layer charging current is no longer considered negligible relative to the faradaic current. Here we consider that film thickness remains constant (i.e., the capacitive component is constant) and scan rate is the only varying parameter in pdif,sol surf . The global CV responses are then of the type shown in Figure 3, where the faradaic waves develop on top of quasi-horizontal capacitive current. We have represented the variations of peak current for an arbitrary value of the RTCd/F2SCmdf parameter (0.1) that measures the relative magnitudes of double layer charging current and faradaic current. Other values of this parameter would simply results in vertical translation of the solid blue curve. The previous remarks concerning the inapplicability of eq 5 also hold in the present case. We now address a different system described by Scheme 2, where P and QH are surface-confined reactants. ZH/Z− is a buffer couple present in the solution, whose diffusion to and from the electrode surface may participate in current control. Proton transfer involving the ZH/Z− couple is assumed to be fast, so as to remain at equilibrium. As detailed in section 2.1, interactions between reactants are described by positive values of the parameter 2(aQ + aP − 2aPQ) = FΔE°/RT. It can be shown (see Supporting Information) that the CV response is governed by a single dimensionless parameter:

Figure 4. Transition between surface reaction control and diffusion control. (a, a′) Competition with film diffusion. (a) CV responses for dif,film psurf = df [Fv /RTDf ]1/2 = 0.1 (black), 0.5 (blue), 1 (red), 2 (green), 3 (magenta), 4 (cyan), 5 (gray), and 10 (orange). (a′) Variation of peak current with the competition parameter: (red) when the double layer charging current is negligible and (blue) in the presence of double layer charging in the case where RTCd/F2SCmdf = 0.1 and constant film thickness is assumed. (b, b′) Competition with solution dif,sol = (Γm/C bZ)[Fv /RTDsol ]1/2 = diffusion. (b) CV responses for psurf 0.5 (blue), 1 (red), 2 (green), 3 (magenta), 5 (gray), and 10 (orange). (b′) Variation of peak current with the competition parameter: (red) when the double layer charging current is negligible and (blue) in the presence of double layer charging in the case where RTCd/F2SCmdf = 0.1 and constant film thickness is assumed.

i p ∝ k1v + k 2 v

(5)

It is easily seen that eq 5 can be rewritten as ip FSCmdf (Fv /RT )

= A1 +

A2 dif,film psurf

where Cb = [Z−] + [ZH] is the total solution bulk buffer − concentration and DZH,Z is the average diffusion coefficient of sol buffer components in the solution. The current−potential response is thus given by an integral equation that can be computed numerically (see Supporting Information for derivation and numerical calculation). Examples of dimensionless CV responses are given in Figure 5. At low scan rates, high buffer concentrations, and small values of Γm (thin films), pdif,sol,H → 0 and a surface wave is surf obtained. In this limit, there is no effect of buffer, and peak current is proportional to scan rate. We are back to the situation analyzed in section 2.1 and Figure 2. Conversely, at high scan rates, low buffer concentrations, and high values of Γm (thick films), pdif,sol,H → ∞ and a diffusion wave is obtained. surf The forward wave is fully controlled by buffer diffusion, and the peak current is proportional to Cb and √ν. The shapes and characteristics of these diffusion waves are, however, not as simple as those of a classical Nernstian Randles−Sevcik wave.8c,12,13 This is illustrated by the black curve in the right

(6)

where A1 and A2 are two constants involving the various parameters that have been introduced earlier. It is immediately apparent that no values of A1 and A2 can be found that would made it possible to fit the variations of peak current with scan rate as shown in Figure 4a′. It follows that analysis of previous pseudocapacitance data based on the application of eq 5 should be re-examined for the sake of avoiding artifacts. 2.3.2. Case 2: Diffusion of Z+ in Solution Is Kinetically Limiting. We address now the case where Z+ transport in the film is fast and competition involves the surface reaction and diffusion of Z+ in the solution. As shown in Supporting Information, the competition between surface reaction behavior and diffusion in solution of the inserting cation is governed by the following parameter: 8654


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Figure 5. Transition between surface wave and diffusion-controlled wave as a function of competition parameter p in the framework of the proton-coupled electron transfer (PCET) mechanism for FΔE°/RT = dif,sol,H ZH,Z− = (Γm/C b)/[Dsol RT /Fv]1/2 = 0 (blue), 1 (red), 1.5 4. psurf (magenta), 2 (green), 4 (orange), 5 (yellow), 10 (gray), and 100 (black) for the case where [ZH]bulk = [Z−]bulk (pH = pKa).

Figure 6. Cyclic voltammetry current−potential responses (E = 1.29 → 0.59 → 1.59 → 1.29 V vs SHE) at 8 V/s of a 39 nm CoPi film in the presence of 0.2 M potassium phosphate at pH = 7.

panel of Figure 5, which shows the current−potential CV response for diffusion-controlled processes in which [ZH] = [Z−] (pH is set at a value equal to the pKa of the acid/base buffer couple). For a given value of the ratio [ZH] = [Z−] (here = 1), the peak heights and the shape of these diffusioncontrolled waves are independent of the magnitude of the interactions. What changes is the potential location of the peaks according to the abscissa of Figure 5 (right panel). The peculiar shape of the reverse scan trace can be explained as follows. At the electrode surface, [Z−] already has a significant value at the start and it accumulates during the first part of the scan. On the reverse scan, the current featuring diffusion is thus larger than during the forward scan. Consequently, QH formed during the forward scan is reoxidized faster and falls rapidly to zero. 2.4. Detailed Analysis of an Illustrative Experimental Example: Electrochemical Charging and Discharging of Phosphate Cobalt Oxide Films. Taken from a series of experiments carried out at various thicknesses and phosphate buffer concentrations, Figure 6 shows an example of CV response to global scanning of the various potential regions of interest for a phosphate cobalt oxide film. As discussed in detail in ref 25, four zones of potential can be delineated. At potentials below ca. 0.8 V versus standard hydrogen electrode (SHE), the film acts as an insulator. As the potential is increased, the film becomes conductive and two surface protoncoupled electron transfer (PCET) waves are observed. At still higher potentials, catalytic O2 evolution occurs. When the concentration of phosphate buffer is decreased to 1 mM, a value just necessary to maintain a stable pH of the bulk solution, regular double layer charging and discharging responses are observed as represented in Figure 7. As expected from a double layer charging response, the plateau current is proportional to the scan rate (Figure 7). Our main point here is that, in the potential region where the cobalt oxide film has passed from the insulating to the conducting regime, it behaves like a standard electronic band conductor (see ref 36 for a detailed discussion of the metallic character of RuO2, a cousin species of our cobalt oxide). A standard electrochemical double layer thus develops at the interface with

Figure 7. Cyclic voltammetry current−potential responses in the presence of 1 mM Pi and 100 mM KNO3 at pH = 7 of CoPi film of various thicknesses (the number on each diagram is the film thickness in nanometers) at various scan rates (in volts per second): 0.02 (green), 0.05 (blue), 0.1 (yellow), 0.2 (red), 0.5 (cyan), 0.05 (blue), 1 (magenta), 2 (gray), 4 (light blue).

the bathing solution. At this stage, it is clear that the capacitance involved in the system is not a pseudocapacitance involving a faradaic process but simply a true capacitance. If it is assumed that one cobalt atom is deposited per electron passed during film electrodeposition, the number of atom moles of cobalt is nCo = (1.04 × 10−8)Ch mol/cm2, where Ch is the deposited charge. Considering that the stoichiometry of the film structure is approximately two oxygen atoms per cobalt38 and not considering the mass of phosphate and water, we obtain a capacitance, defined as the double layer capacitance per unit of mass, of ca. 200−300 F/g. What happens now if we turn on the faradaic reactions by adding more phosphate buffer in the solution? Figure 8 shows linear scan current−potential responses, observed in the 1.3− 0.6 V vs SHE potential range at various film thicknesses, as a function of phosphate concentration and scan rate. Two reversible and peak-shaped waves appear. These involve the following faradaic PCET reactions:25 8655


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concentration is high. Under these conditions, the effect of phosphate diffusion is negligible. The wave is then a surface faradaic wave whose broadness is indicative of interactions between the CoIV and CoIII reactants. As shown in section 2.1, this description is equivalent to a square distribution of standard potentials with FΔE°CoIV/III/RT = 4 and Γm = 8 × 10−9 mol/cm2.25 The CoIII/II faradaic wave is more difficult to analyze because the local buffer concentration is modified by scanning over the preceding Co IV/III wave, except at the highest buffer concentration and the lowest scan rate, where it can be seen that both waves tend to have the same shape (Figure 8). We may thus conclude that these two waves involve true faradaic reactions that occur on top of a large true capacitive double layer charging response. Both processes may be used for charge storage, and it is worth noting that a large buffer concentration allows for more than doubling the apparent film capacitance if it is defined from total charge stored in the 1.3− 0.6 V vs SHE potential range.

3. CONCLUSIONS AND PERSPECTIVES Faradaic reactions in batteries and the double layer charging processes in electrochemical capacitors are unambiguously distinguished by their electrochemical signatures (e.g., cyclic voltammetry). Double layer charging/discharging processes exhibit quasi-rectangular CV responses with a current displaying proportionality to scan rate. The notion of pseudocapacitance has been introduced to characterize systems involving faradaic processes that simultaneously display the same CV signature as double layer charging/discharging processes. Superposition of overlapping redox couples has been invoked to explain this surprising congruence. We have investigated this possibility after having shown that this approach is closely equivalent to taking into account the more physically meaningful occurrence of concentrationdependent activity coefficients resulting from interactions between reactants. The result of the ensuing theoretical analysis is that, regardless of the application of one or the other of these two approaches, the faradaic CV responses never take a quasirectangular CV shape proportional to scan rate that could be similar to that of a double layer charging/discharging response. In all cases, peak-shaped responses are obtained, which are more or less thickened by interactions between reactants. It follows that whenever a quasi-rectangular CV response proportional to scan rate is observed, such reputed pseudocapacitive behaviors should in fact be ascribed to truly capacitive double layer charging. In practice, double layer charging and faradaic responses are often superimposed, showing more or less broadened peaks on top of a quasirectangular CV response proportional to scan rate. Such CV responses are observed in the abundant metal oxide literature, including the classic examples RuO2 and MnO2. Although available data in the field confirm our analysis qualitatively, double layer charging and faradaic components are, in most cases, difficult to deconvoluate owing to uncontrolled kinetic and ohmic drop effects, as well as incorrect treatments of the scan-rate dependence of the CV response. Because the latter issue is a clue to the possible contribution of diffusion to the CV response, we developed a rigorous analysis of the problem, leading to a quantitative evaluation of the respective importance of interfacial reactions and diffusional transport in the global faradaic response. Two simple cases were treated in this regard. In one of these, diffusionlike cation-coupled electron hopping

Figure 8. Linear scan voltammetry (i/v) potential responses (E = 1.29 → 0.59 V vs SHE) of CoPi film as a function of thickness (columns from left to right: 39, 78, and 156 nm); scan rates (in volts per second) 1 (blue), 2 (red), 3 (green), 4 (magenta), 5 (yellow), 6 (gray), 7 (cyan), and 8 (orange); and [phosphate] (rows from top to bottom: 1, 20, 50, 100, and 200 mM) at pH = 7. The gray rectangle represents the capacitive current as obtained from data at 1 mM phosphate.

CoIV O + PO4 H 2− + e− ⇌ CoIIIOH + PO4 H2 −

CoIIIOH + PO4 H 2− + e− ⇌ CoIIOH 2 + PO4 H2 −

At a given scan rate and phosphate concentration, the peak currents increase with film thickness, as expected for a porous material with redox couples at the edges of constituting oxidized cobalt clusters.24 At a given film thickness and buffer concentration, the variation of the CoIV/III peak with scan rate is between proportionality to v and to v1/2, indicating that buffer diffusion interferes as described in section 2.3.2. For the thinnest film (39 nm) and lowest scan rate (1 V/s), the CoIV/III peak tends to become proportional to scan rate when the buffer 8656


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*(J.-M.S.) E-mail [email protected].

accompanies the interfacial reaction as in insertion metal oxide electrodes. In the other case, cation diffusion in the solution accompanies interfacial electron transfer. The use of the scan rate, v, as a diagnostic criterion of the respective role of diffusion (proportionality to v1/2) and surface reaction (proportionality to v) has been clarified in both cases. A more complicated case is when diffusion of a buffer in solution interferes and interactions between surface reactants are taken into account. It served as a theoretical prelude to quantitative testing of our analyses with phosphate cobalt oxide films. Although these films are not among the most efficient metal oxide charging devices, they are suitable for our purpose. One of their useful features is indeed that the relative importance of faradaic processes can be modulated by adjusting the phosphate buffer concentration. Our conclusions do not invalidate the numerous experimental studies carried out under the pseudocapacitance banner, but rather they provide a more physically meaningful framework for their interpretation. On the bases laid by this clarification, an important future task will be to analyze the factors that govern the rapidity of the charge/discharge process. This implies taking into account that one is dealing with systems involving mesostructures, sets of nanoparticles, interfacial kinetics, etc., for which the mastering of diffusion and ohmic loss problems is currently underway.39

ORCID

Jean-Michel Savéant: 0000-0003-1651-3153 Funding

Partial support from the Agence Nationale de la Recherche (ANR CATMEC 14-CE05-0014-01) is acknowledged. Notes

The authors declare no competing financial interest.

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(1) Winter, M.; Brodd, R. J. What Are Batteries, Fuel Cells, and Supercapacitors? Chem. Rev. 2004, 104, 4245−4269. (2) Simon, P.; Gogotsi, Y. Capacitive Energy Storage in Nanostructured Carbon−Electrolyte Systems. Acc. Chem. Res. 2013, 46, 1094−1103. (3) Tsai, W.-Y.; Taberna, P.-L.; Simon, P. Electrochemical Quartz Crystal Microbalance (Eqcm) Study of Ion Dynamics in Nanoporous Carbons. J. Am. Chem. Soc. 2014, 136, 8722−8728. (4) Pean, C.; Daffos, B.; Merlet, C.; Rotenberg, B.; Taberna, P.-L.; Simon, P.; Salanne, M. Single Electrode Capacitances of Porous Carbons in Neat Ionic Liquid Electrolyte at 100°C: A Combined Experimental and Modeling Approach. J. Electrochem. Soc. 2015, 162, A5091−A5095. (5) Enterría, M.; Figueiredo, J. L. Nanostructured Mesoporous Carbons: Tuning Texture and Surface Chemistry. Carbon 2016, 108, 79−102. (6) Tanimura, A.; Kovalenko, A.; Hirata, F. Molecular Theory of an Electrochemical Double Layer in a Nanoporous Carbon Supercapacitor. Chem. Phys. Lett. 2003, 378, 638−646. (7) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; John Wiley & Sons: 2001. (8) Savéant, J.-M. Elements of Molecular and Biomolecular Electrochemistry: An Electrochemical Approach to Electron Transfer Chemistry; John Wiley & Sons, Hoboken, NJ, 2006; DOI: 10.1002/0471758078. (a) Pages 14−16 and 353−357. (b) Pages 3−5. (c) Chapter 1. (9) Delahay, P. Double Layer and Electrode Kinetics; Wiley: New York, 1955; Chapt. 7. (10) Laviron, E. Influence of Adsorption of Depolariser or of a Product of Electrochemical Reaction on Polarographic Currents. 2. Theoretical and Experimental Investigation of Behaviour of Methylene Blue by Linear Potential Sweep Voltammetry. Bull. Soc. Chim. Fr. 1967, 3717−3721. (11) Laviron, E. Voltammetric Method for the Study of Adsorbed Species. In Electroanalytical Chemistry: A Series of Advances, Vol. 12; Bard, A. J., Ed.; Dekker: New York, 1982. (12) Randles, J. E. B. A Cathode Ray Polarograph. Part II. The Current-Voltage Curves. Trans. Faraday Soc. 1948, 44, 327−338. (13) Sevcik, A. Oscillographic Polarography with Periodical Triangular Voltage. Collect. Czech. Chem. Commun. 1948, 13, 349−377. (14) Matsuda, H.; Ayabe, Y. Zur Theorie der Randles-Sevcikschen Kathodenstrahl-Polarographie. Z. Elektrochem. 1955, 59, 494−503 ( http://onlinelibrary.wiley.com/doi/10.1002/bbpc.19550590605/pdf). (15) Conway, B. E. Transition from “Supercapacitor” to “Battery” Behavior in Electrochemical Energy Storage. J. Electrochem. Soc. 1991, 138, 1539−1548. (16) Conway, B. E.; Birss, V.; Wojtowicz, J. The role and utilization of pseudocapacitance for energy storage by supercapacitors. J. Power Sources 1997, 66, 1−14. (17) Conway, B. E. Electrochemical Capacitors: Scientific Fundamentals and Technological Applications; Kluwer Academic/Plenum Publishers: New York, 1999; DOI: 10.1007/978-1-4757-3058-6. (18) Brousse, T.; Belanger, D.; Long, J. W. To Be or Not To Be Pseudocapacitive? J. Electrochem. Soc. 2015, 162, A5185−A5189. (19) Brousse, T.; Toupin, M.; Dugas, R.; Athouël, L.; Crosnier, O.; Bélanger, D. Crystalline MnO2 as Possible Alternatives to Amorphous Compounds in Electrochemical Supercapacitors. J. Electrochem. Soc. 2006, 153, A2171−A2180.

4. EXPERIMENTAL SECTION 4.1. Materials. All solutions were prepared with Milli-Q water. Co(NO3)2·6H2O (99.8%), KOH (≥85%), KNO3 (≥99%), and KH2PO4 (≥99%) were purchased from Sigma and used as received. The 1 mm diameter Pt electrode (purchased from Metrohm) was polished with aluminum paste on a micropad cloth. After polishing, the electrode surface was cleaned electrochemically by successive potential sweeps in ∼0.2 M HNO3 (from 0.25 to −0.30 V vs SHE, 1 V/s, ×10) and then anodized at 1.44 V versus SHE in 100 mM phosphate buffer solution (pH = 7) for ∼5 min immediately prior to film deposition. This cleaning protocol was repeated before each film preparation. CoPi films of different thicknesses were deposited onto the Pt working electrode as previously described. 4.2. Electrochemical Methods. Cyclic voltammetry was performed with an Autolab potentiostat using a 1 mm diameter Pt working electrode, a platinum wire counter electrode, and a saturated calomel electrode (SCE) reference electrode. All potentials were converted from SCE to SHE by E(SHE) = E(SCE) + 0.241 V. Between experiments, the cell was held at a constant potential of 1.09 V versus SHE to prevent the slow film dissolution that occurs at open circuit potential. Unless otherwise specified, all solutions were adjusted to pH = 7 and contained 100 mM KNO3 in addition to the phosphate buffer, Pi. In all CV scans, the cell resting potential immediately prior to scan initiation was 1.09 V versus SHE. Positive feedback compensation was used to minimize the ohmic drop between the working and reference electrodes. In all cases, excellent reproducibility was observed between different films under all conditions reported.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsami.6b14100. Glossary of symbols; additional text and equations expanding upon section 2.1, section 2.3, Scheme 1, and Scheme 2 (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Authors

*(C.C.) E-mail [email protected]. 8657


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Cobalt-Based Oxygen-Evolving Catalyst. J. Am. Chem. Soc. 2013, 135, 6403−6406. (39) Andrieux, C. P.; Costentin, C.; Di Giovanni, C.; Savéant, J.-M.; Tard, C. Conductive Mesoporous Catalytic Films. Current Distortion and Performance Degradation by Dual-Phase Ohmic Drop Effects. Analysis and Remedies. J. Phys. Chem. C 2016, 120, 21263−21271.

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