Supercapacitors

## **Chapter 2**

## Redox Mediated Electrolytes in Electrochemical Capacitors

*Paulina Bujewska, Przemysław Galek, Elżbieta Frąckowiak and Krzysztof Fic*

## **Abstract**

Electrochemistry is strongly related to redox reactions. Charge transfer processes are used for the current generation in all electrochemical cells. Nowadays, redox reactions are still of evitable importance for energy storage/conversion technology. For instance, the charge and discharge of batteries exploit redox reactions. Moreover, these processes can also be used to improve the operating parameters of other energy storage devices like electrochemical capacitors. Although, in principle, the energy in electrochemical capacitors is stored in an electrostatic manner (by electrical double-layer formation), the redox reactions introduce an additional charge and improve the energy of these systems. This chapter presents the principles of electrochemical capacitors' operation and provides comprehensive insights into this technology with special attention focused on hybrid systems, exploiting the redox activity of the electrolytic solution.

**Keywords:** energy storage devices, electrochemical capacitor, redox-active electrolytes, aqueous electrolytes, organic electrolytes, ionic liquids

## **1. Introduction**

Growing demand for energy in all possible forms (mobility, heat, electricity) induced intensive research on various energy harvesting and storage systems. Today, it is clear that fossil fuels are no longer a reasonable choice for further society development. Various reasons, such as environmental pollution, depletion of natural resources, and remarkable climate changes, stimulated intensive research on sustainable solutions for energy harvesting and storage. In this context, numerous technologies are known for sourcing the energy in a "green" manner (like photovoltaics, wind turbines, flywheels); however, this energy must be somehow stored to be further used when needed.

Electrochemical energy conversion and storage systems are one of the most common solutions used every day by almost everyone—at home, at work, or in the car. This is true that well-known Li-ion batteries allowed the world "to move" and made our lives more "mobile". Nevertheless, these are not the only ones that have been recently developed and used. Despite the high amount of energy stored in batteries, their power density is still not enough to eliminate other technologies. Furthermore, their typical redox-based charge storage mechanism makes their lifetime short (counted in thousands of cycles) and thus less resource-effective. Electrochemical

capacitors, with their high-power density and moderate energy, cyclability counted very often in millions of cycles and much safer chemistry in the cell, appear to be an interesting technology that could serve as a standalone system or greatly accompany the battery.

## **2. Electrochemical capacitors: definition, construction, types**

Conventional capacitors are composed of two flat, non-porous plates (electrodes) separated by a dielectric material. These devices are characterized by low energy density, limiting their application [1]. In 1957, a new group of capacitors, called electrochemical capacitors (ECs), super- or ultracapacitors, emerged. It must be here pointed out that only the "electrochemical capacitor" term should be used for scientific purposes, as other names (supercapacitors, ultracapacitors, etc.) refer to commercial products. Furthermore, "electrochemical capacitors" are often confused with "electric double-layer capacitors (EDLCs)". In fact, EDLCs are always ECs; however, this term is reserved only for the systems exploiting the double-layer charging/ discharging process, thus, the mechanism is entirely electrostatic, while ECs could also exploit redox-based processes in the charge storage (like hybrid systems).

Unlike conventional capacitors, in ECs, the electrodes of highly developed surfaces are used. Such electrodes allow higher capacitance to be reached and, in consequence, the energy accumulated increases while their superior power is maintained [2].

It is worth noting that besides ECs, there are many energy storage devices and their application depends on the performance parameters. Therefore, these properties, i.e., energy and power, are crucial from a practical point of view. The so-called Ragone plot (**Figure 1**) is the best way to compare various systems' performance [4].

It can be noticed that the electrochemical capacitors demonstrate the properties between conventional capacitors and batteries—the specific power is very high, however, slightly lower than in the case of "dielectric" capacitors, and the specific energy is significantly higher—but still moderate if compared with the batteries (especially

#### **Figure 1.**

*Ragone plot presenting the performance parameters (energy and power) of different energy storage/conversion devices [3].*

commonly used Ni/MH, Li-ion and Li-primary ones). Besides the tremendous power of ECs that allows them to be charged and discharged very quickly, these devices are getting more and more attention because of their long lifetime and safe/reliable use [5]. For these reasons, ECs are applied in the automotive industry—for instance, in regenerative braking, start-stop systems, or track control devices. Nonetheless, the energy density of these devices needs to be increased, as the volume or weight of the device must be reduced.

As already mentioned, ECs consist of two porous electrodes of highly developed surface area. The electrodes are very often made of carbon materials (especially activated carbons) due to their good conductive properties, high availability (abundance) as well as relatively low price [6]. During the ECs operation, the electrodes are polarized positively (+) and negatively (). An insulator separates them to prevent short circuits. These components are immersed in an electrolyte, playing the role of ion source and carrier (**Figure 2a**). When charging the cell, positively charged ions (cations) are adsorbed on the () electrode surface, while negatively charged ions (anions) are adsorbed on the (+) electrode surface. An electrical double-layer is formed at the electrode/electrolyte interface during this process. For this reason, ECs are also called electric double-layer capacitors—EDLCs (**Figure 2b inset**). The opposite process (discharge) results in the desorption of ions from the electrode surface vicinity to the electrolyte volume (**Figure 2c**) [7, 8].

Carbon materials can be enriched with surface functional groups, heteroatoms like oxygen or nitrogen, and transition metal oxides like MnO2. Moreover, carbon/electrically conductive polymer (e.g., PANI, PPy, PEDOT) composites can be synthesized and used as electrodes for ECs. These materials are classified as pseudocapacitive ones [9]. The charge storage mechanism in such devices can be described as quick, continuous faradaic reactions occurring with no phase change in the electrode material. The cells operating with these materials are very often called asymmetric or pseudocapacitance-based ECs (**Figure 2b**). One should restrain from using the "pseudocapacitor" term, as the pseudocapacitance concerns the electrode, not the

**Figure 2.** *Electrochemical capacitor: (a) construction, (b) types and (c) principle of operation.*

#### **Figure 3.**

*Comparison of the voltammetric responses of a positively (+) and negatively (*�*) polarized electrode of the electric double-layer capacitor (solid line) and a redox (hybrid) capacitor (dashed line) [10].*

system. If redox reactions occur on both electrodes, the system should rather be considered as a battery.

ECs incorporating pseudocapacitive materials may suffer from shorter cycle life, due to unstable behavior of the functional groups during long-term tests and chemical and mechanical composites degradation. Moreover, the cost of such materials exceeds the cost of non-modified activated carbon and impacts the final price of the cell. Thus, another solution was proposed to increase the capacitance, causing an increase in the energy of the ECs—i.e., electrolytes demonstrating redox activity (redox ECs, **Figure 2b**).

Generally, the redox processes in the batteries are attributed to the electrode material, ensuring high charge storage capacity. However, solid-state diffusion remarkably impacts the power capability. Shifting the redox processes to electrolytic solution remarkably diminishes the mass-transfer limitations and allows the power of electrostatic interactions to be almost maintained.

The operating potentials of each electrode in symmetric EDLCs are comparable. For instance, when ECs are investigated with cyclic voltammetry, the curves of rectangular shape are recorded (**Figure 3**; solid lines), since the capacitance does not depend on the potential.

In the case of galvanostatic charge/discharge, the curves are triangular [11]. Obviously, it is possible to notice potential shifts (very often negligible) that originate from matching cations/anions with the pore diameter of the electrode material. The capacitance of the system (Ccell) can be calculated based on Eq. (1) because two electrodes that store the energy at the electrode/electrolyte interface are considered as two capacitors in series [10].

$$\frac{1}{C\_{cell}} = \frac{1}{C^{+}} + \frac{1}{C^{-}} \tag{1}$$

Assuming the capacitance values of both electrodes in symmetric cell are comparable (*C*<sup>+</sup> ≈ *C*� = *Cele*), Eq. (1) can be transformed to Eq. (2):

*Redox Mediated Electrolytes in Electrochemical Capacitors DOI: http://dx.doi.org/10.5772/intechopen.104961*

$$\mathbf{C}\_{cell} = \frac{\mathbf{C}\_{ele}}{2} \tag{2}$$

The specific energy for the EDLCs (*EEDCL*) can be calculated from Eq. (3):

$$E\_{\rm EDCL} = \mathbf{0.5CV^2} \tag{3}$$

where *C* can be calculated from Eq. (4):

$$C = \frac{Q}{m\Delta V} \tag{4}$$

For accurate calculations, it is necessary to consider the ohmic drop for Δ*V* calculation [10].

In the case of ECs operating in redox-active electrolytes, the potential range of each electrode can significantly differ, as presented in **Figure 3** (dashed lines). It is seen that one electrode demonstrates capacitive character, typical of EDL formation, with constant capacitive current recorded; at the same time, the positive electrode demonstrates a very high current response with a narrow potential range. This suggests high capacity, accumulated in a narrow potential range, typical of the redox process. In the galvanostatic charge/discharge technique, the redox activity is seen as a *plateau* on the *E* = *f*(*t*) plot [11].

For the cells' performance characterization, the specific energy (*E*) should be calculated from the galvanostatic charge/discharge profile, with applied current *I* and change in the voltage (*V*) over the time (*t*), recalculated per active mass (*m*) of both electrodes:

$$E = \frac{1}{3600 \cdot m} \left[ \text{VI} dt \right] \tag{5}$$

Power capability needs to be calculated as well for the full characterization of the investigated cells. It is directly related to the system's energy, according to Eq. (6):

$$P = \frac{E}{\Delta t\_{disch}}\tag{6}$$

where Δ*tdisch* is the discharge time at which the energy is released.

For more detailed information and characterization techniques, comprehensive literature reports are published [9, 12, 13].

It must be clearly stated that the energy and power of the devices should be expressed per mass of the cell components and must not be calculated for the single electrode. However, on the laboratory scale, when the electrolyte is in great excess, only the mass of the electrolyte confined in the pores should be considered. The other possibility is to normalize these values per volume of the device's components. All the presented methods of cells characterization is correct, but the author needs to comment on how the calculations were made [10–12].

## **3. Redox-mediated electrolytes**

As mentioned, the redox-active electrolyte in EC allows the cell performance to be significantly improved. It is necessary to use the electrodes made of electrically

#### **Figure 4.**

*Redox couples with their reduction potentials [18]. Redox couples marked in red are stable in acidic conditions, those marked in green are stable in neutral solutions and the one in blue is stable in alkaline electrolytes.*

conductive material to make the electron flow from the electrode to the electrolyte possible [14–17].

There are many redox couples with well-defined and stable redox activity that can be used as additives for electrolytic solutions. The most popular ones, with their reduction potentials (expressed *vs.* normal hydrogen electrode; NHE), are presented in **Figure 4** [18].

Depending on the cell construction, electrode material used, potential application, and expected operating performance, one can select which redox couple is suitable for EC that meets the requirements. In the case of aqueous-based systems, there are additional issues that need to be taken into account. First of all, at too high or too low potentials, water is decomposed, so oxygen and hydrogen evolution can be observed, respectively. These reactions are considered harmful for the cell because (i) the solvent should not be decomposed, (ii) evolving gases can block the electrode porosity, (iii) the highly active oxygen causes the irreversible electrode oxidation and its degradation, and (iv) corrosion of the current collectors remarkably affects the cell lifetime. Therefore, the potential of the chosen redox couple should preferably be between hydrogen (HEP) and oxygen (OEP) evolution potential.

The second issue is related to the electrolyte pH. Both HEP and OEP are pHdependent—when the solution pH increases, these potentials are shifted toward lower potentials [19]. It is, thus, possible to slightly adjust the HEP and OEP by regulating the electrolyte pH. However, one should keep in mind that the potentials of some redox couples are also pH-dependent, so with the pH change, their potential will also change. Moreover, the stability of redox couples also depends on the solution pH.

Redox-active electrolytes are grouped in a way similar to the types of electrolytes. Hence, they can be divided into two main groups—aqueous and nonaqueous ones [11, 20].

### **3.1 Aqueous redox-active electrolytes**

Aqueous solutions, despite their limited operating voltage related to the theoretical water decomposition above 1.23 V, are very attractive electrolytes for ECs due to their price lower than for nonaqueous electrolytes and the possibility of the cell manufacturing in an ambient atmosphere. Moreover, the impact of water-based solutions on the environment is rather negligible. These solutions are also characterized by high conductivity and low viscosity. The main drawback of the ECs operating in

redox-active electrolytes is moderate cycle life related to the efficiency of the redox reactions and possible side reactions [21, 22].

In general, it seems beneficial to combine more than one redox additive in one electrolyte. If the ratio between different redox species is well-optimized, the energy reached in such cells is higher than reported for the single redox couple [23, 24].

Aqueous redox-active electrolytes can be divided into three groups: cationic, anionic, and neutral electrolytes, due to the charge of the redox-active ion. It is worth mentioning that the redox ions in cationic and anionic electrolytes contribute to the EDL formation, whereas in neutral electrolytes redox species quite often do not participate in this process [11].

## *3.1.1 Cationic aqueous redox-active electrolytes*

Cationic redox electrolytes can be divided into three groups: lanthanides, transition metals, and organic species [18, 25–31]. The general requirement is that the solubility of these species should be possibly high and their standard potential should be close to HEP, as their activity is expected at the negative electrode [18].

Cerium, which belongs to lanthanides, was introduced to the acidic solution [11, 27]. However, standard redox potentials of lanthanides (+1.6 V vs. NHE of Ce3+/ Ce4+ redox couple) being higher than OEP definitely limits their application.

The second group—transition metals like Zn, Sn, Mn, Fe, Ni, Cu, include a solid phase in the solution of neutral or acidic pH. The cations are reduced at relatively low potential, between 0.762 V and +0.337 V vs. NHE. However, still irreversible hydrogen evolution reaction can occur during the metal electrodeposition in aqueous solutions. Although in general hydrogen evolution reaction is considered parasitic or unwanted, it is possible to store hydrogen reversibly in the electrode porosity—it is necessary to use microporous electrodes for this purpose. Moreover, the addition of halide ions to the electrolytic solution can be beneficial—halide anions will block the carbon and its active sites preventing hydrogen reactions [32, 33]. Finally, metal electrodeposited on the electrode can affect the specific surface area of the electrode and worsen the performance stability of the system. It is also possible to avoid solidstate metal deposition on the electrode, by applying redox couples dissolved in the liquid state, like Fe2+/Fe3+, Cu+ /Cu2+ [26, 34].

Viologen di-cations can be included in the organic cationic additives. Moreover, these species are characterized by fast redox kinetics and high reversibility [35, 36]. It was found that 1,10-dimethyl-4,40-bipyridinium cation (MV2+) is strongly attracted to the electrode surface. However, after reduction to MV+ , the physical interaction between these species and the electrode can be even stronger [37]. This may be beneficial to reduce self-discharge, which is caused by redox shuttling.

As the cations are supposed to be attracted to the negatively polarized electrode, redox reactions originating from cationic additives are mostly at the negative side. However, the synthesis of carbon material exhibiting the affinity to cations and application of such an electrode as the positive one in ECs is also reported [26, 38–40]. It is worth noting that not only carbon materials can be functionalized—in fact, but various polymers can also be enriched with cationic (or anionic) functional groups.

The systems operating in redox-active electrolytes with transition metals as active species need to be assembled with ion-selective membranes as separators. These membranes can mitigate the self-discharge and leakage current which are relatively high for such systems [23, 39]. Nevertheless, the application of viologens (organic

molecules) as a redox additive to the electrolytic solution can also decrease selfdischarge without the necessity of ion-selective membrane employment. It is caused by viologens strong adsorption at the porous electrode surface [18, 28].

The main disadvantage of using viologens is their limited solubility and large size of the molecule that can negatively influence the ECs performance [32, 41], especially because of mass-transport issues.

### *3.1.2 Anionic aqueous redox-active electrolytes*

The anionic redox-active electrolytes contain halides (iodide [14, 42–44], bromide [4, 45], pseudohalides (thiocyanate [41], selenocyanate [46]), organometallic complexes (ferricyanide and, ferrocyanide [21, 47–53]) and organic anion—like indigo carmine [54].

In the case of halide and pseudohalides-based electrolytes, a well-defined redox response is recorded at the positively polarized electrode. They are characterized by strong adsorption at the electrode surface. Hence, the self-discharge of the cell operating in such electrolytes is relatively low and the application of an ion-selective membrane is not needed. Moreover, halides can be coupled with metal ions deposition reaction, especially Zn/Zn2+, and viologen redox couple [30, 31, 55], however, such systems are no longer typical capacitors. To avoid metal dendrites formation, some additional components should be used, like dendrite suppression or nanoporous separators [56–59].

The standard potentials of bromide and iodide reactions are similar; however, the bromides demonstrate slightly higher values [60]. It can be beneficial for reaching higher energy of the ECs, as the operating voltage might be shifted toward higher values. Nonetheless, bromide solutions are toxic, so for safety, it is favorable to use iodide-based solutions. Also due to the high standard potential of Br�/Br2, close to oxygen evolution potential, the electrolyte decomposition can be difficult to control and corrosion on current collectors can be observed [31]. Iodide-based ECs are widely described in the literature. These systems are characterized by stable operation even during long-term experiments [32, 61, 62].

Pseudohalides solutions exhibit similar electrochemical behavior to halide solutions when used as electrolytes in ECs but self-discharge is definitely more pronounced. Thiocyanates-based solutions are especially interesting for ECs application due to their higher maximum operating voltage than selenocyanate-based electrolytes. Moreover, the energy and power of such systems are comparable to those reached in iodide-based electrolytes, but their lifespan is still limited [41].

Organometallic-based electrolytes (ferricyanide- or ferrocyanide-based solutions) ensure the promising performance of the ECs. The main drawback of these electrolytes is high self-discharge, seen as low efficiency, especially at low current loads. Therefore, ion-selective membranes are very often used to limit redox shuttling [48, 63].

### *3.1.3 Non-ionic aqueous redox-active electrolytes*

Even in aqueous-based systems, organic redox-active additives can be used. For instance, hydroquinone (HQ), anthraquinone [64–66], catechol (an isomer of benzoquinone), rutin [67], p-phenylenediamine [68], and conducting polymers [69, 70] (if soluble in water) are popular neutral electroactive species added to the electrolytic solutions. To enable redox reaction with proton transfer, the use of supporting electrolytes is necessary. For this reason, acid solutions (H2SO4) are used as a source of

protons. As a consequence, the maximum operating voltage of the ECs operating in an acidic medium is limited to �1 V, and, because of corrosion issues, the use of gold or other noble metal current collectors is necessary.

As the representative reaction, the reduction of benzoquinone to hydroquinone (Q/HQ) is presented in Eq. (7).

Moreover, the conductivity of the electrolytic solutions with organic molecules can be diminished. Therefore, additional ionic species are very often introduced (like neutral salts—KNO3 or alkaline KOH [71]); hence, the formation of EDL can be more efficient. These systems are also characterized by considerable self-discharge related to the movement of neutral molecules between the polarized electrodes. To reduce self-discharge and increase the efficiency of the charging and discharging processes, the use of an expensive proton exchange membrane is recommended, which significantly increases the price of ECs [72]. The cells operating in the electrolytes with polymeric additives (i.e., sulfonated polyaniline or p-nitroaniline) also required the use of cheaper membranes. It is possible to use a semipermeable membrane that allows the movement of protons and supporting ions like SO4 <sup>2</sup>�. The drawback of such electrolytes is the solubility of the polymeric molecules—when the concentration of electroactive molecules is relatively low, the capacity of the cell is also limited [69]. Therefore, it is necessary to investigate the ECs with new polymer-based electrolytes to develop these systems and reach satisfactory operating parameters.

## *3.1.4 Cationic-anionic electrolytes*

As cationic additives exhibit redox activity at the negatively polarized electrode and anionic additives at the positively polarized one, they can be combined, giving significant performance improvement. These redox couples should be carefully selected because they must be stable and soluble under the same conditions. Otherwise, it would be necessary to use more expensive separators/membranes and the assembly process would be more complex [18]. ECs operating in the electrolyte containing viologen cation and halide anion were tested. In the case of the electrolyte with MV2+ and I� redox-active species during cell charging, an irreversible capacitance loss was noticed. It was caused by precipitate formation (MX•<sup>+</sup> –I �) [73]. When the iodide was replaced by bromide (the anion of higher standard potential) the processes were reversible, and higher energy was reached. However, because of the high potential needed for Br�/Br3 � activity, the signs of corrosion were observed. MVCl2/KBr-based cells suffer from a relatively high self-discharge, which was more pronounced than for halide-based electrolytes, suggesting that MV species are, mostly, responsible for this voltage loss. Therefore, other viologen was used—1,10 diheptyl-4,40-bipyridinium dibromide (HVBr2), resulting in lower self-discharge. Probably, not only stronger adsorption of HV2+ cation was the reason for the lower self-discharge but also these cations were immobilized due to the precipitate formation within the carbon electrode [74]. The optimization of redox-active species concentration, choice of the appropriate counter anion/cation for redox-active cation/

anion, respectively, and experimental conditions optimization is definitely more complex and time-consuming than for one active component within the electrolytic solution. However, taking into account the significant improvement of the energy stored in the EC operating in the redox-active electrolytes, it is still worth discovering the potential of this field.

### **3.2 Redox-mediated nonaqueous electrolytes**

Commercially used ECs very often employ nonaqueous electrolytes (organic ones) despite the fact, that they cannot be considered environmentally friendly solutions, because of the necessity of toxic solvents use—like acetonitrile or propylene carbonate. However, they have a few advantages that make them more attractive for ECs construction: wider electrochemical window (up to 3.8 V [75, 76]) which allows higher energy to be stored, and long cycle life. On the other hand, there are ionic liquids called "green solutions", that can be also used in ECs but they are relatively expensive.

### *3.2.1 Organic electrolytes with redox activity*

In the organic electrolytes, conductive salts like tetraethylammonium tetrafluoroborate (TEABF4), lithium bis(trifluoromethanesulfonyl)imide (LiTFSI), 1 ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (EMImTFSI), lithium hexafluorophosphate (LiPF6), are dissolved in acetonitrile (ACN) or propylene carbonate (PC), which are the most popular solvents for ECs application [22, 77–81]. As already mentioned, organic electrolytes allow the ECs to operate at higher voltages than aqueous-based electrolytes do [82] and they provide higher power than the systems with ionic liquid (IL) electrolytes [11, 83] due to the higher ionic conductivity of organic electrolytes.

The maximum voltage of reported organic-based cells is 2.5 V, when IL (1-ethyl-3 methylimidazolium ferrocenylsulfonyl-(trifluoromethylsulfonyl)-imide, [EMIm] [FcNTf]) in ACN [76] and p-phenylenediamine additive to lithium perchlorate LiClO4 in ACN [84] were used as electrolytes. The mixture of microporous carbon with carbon black and graphite was used as the electrode material. However, there are also other materials that can be used, for instance two-dimensional titanium carbide (MXene) [81].

Organic electrolytes exhibiting redox activity are not as popular as aqueous electrolytes. Therefore, there is a gap in this field of study as there are many possible redox additives that could be employed for organic electrolytes [85].

## *3.2.2 Ionic liquids*

Ionic liquids (ILs) are compounds composed entirely of ions—bulky, usually asymmetric organic cation and anion (weakly coordinating) that can be both organic and inorganic [86, 87]. As they are ionic conductive, there is no need to use additional solvents. They are characterized by high electrochemical stability, ensuring a high voltage window (>3 V) and high thermal stability [88, 89]. It is possible to introduce redox additives to IL, for example by incorporating metal ions (Cu2+ added in the form of copper chloride to [EMIm][BF4] [40], neutral redox molecules [90, 91] (HQ added to [TEA][TFSI] [92]) or sulfates (SnSO4 and VOSO4 [23]).

However, ILs themselves can also exhibit redox activity if an anion of IL is electroactive. Hence, such an electrolyte can be called redox-active IL. To observe

## *Redox Mediated Electrolytes in Electrochemical Capacitors DOI: http://dx.doi.org/10.5772/intechopen.104961*

effective and beneficial redox contribution to ECs charge/discharge, a high concentration of electroactive species needs to be ensured. Electrolyte composed of two ILs— [EMIm][BF4] and [EMIm]Br, where the latter one is a redox additive (1 mol L<sup>1</sup> ) to the former one, was used in microporous electrodes-based EC. The operating parameters were significantly improved due to the bromide activity (the specific energy was almost twice higher if compared to the [EMIm]BF4-based system, where only EDL formation is assumed, and the Coulombic efficiency was 100%) [83]. Moreover, the leakage current was reduced, probably due to strong adsorption of halide on the positively charged carbon electrode, described for aqueous-based cells [18, 32].

Biredox ILs can also be used as electrolytes for ECs. The idea arises due to the potential balancing issue when additives, like metal ions, HQ, or redox-active anions, are introduced to the system operating with microporous carbon electrodes [75]. The cation of IL ([BMIm][TFSI]) was functionalized with AQ, whereas the anion was functionalized with 2,2,6,6-tetramethylpiperidinyl-1-oxyl (TEMPO) molecule. The energy density of such a system was definitely higher than for IL with redox additive as an electrolyte but the specific power and lifetime were rather moderate.

The application of redox-active IL as electrolytes in ECs is a promising strategy to increase the specific energy of the systems. However, one should take into account that the price of such devices is relatively high. Moreover, the power performance and the lifespan of the ECs operating in ILs should be improved.

### *3.2.3 Redox-mediated gel electrolytes*

Gels are characterized by very good stability (both chemical and mechanical) and they can be made of eco-friendly materials [93]. They can be successfully applied as electrolytes (based on aqueous solutions or ILs) for ECs [61, 94, 95]. Gel electrolytes were introduced to ECs to reduce their self-discharge [96] and enable the development of flexible devices, where liquid electrolytes would expose the cells to leaks. Redox mediators can be introduced to the gel electrolytes and increase both ionic conductivity and capacitance of the ECs [97]. For instance, when indigo carmine was added to the gel electrolyte based on polyvinyl alcohol (PVA) and sulfuric acid, the ionic conductivity increased by almost 190% [98]. Moreover, the lifetime of the devices can be prolonged. Redox-active compounds, like 1-butyl-3-methylimidazolium iodide and bromide (BMImI, BMImBr) [99, 100], 1-anthraquinone sulfonic acid sodium [101], 1,4 naphthoquinone [102], including indigo carmine [98] and FeBr3 [103], can be incorporated into gel structure. BMImBr with Li2SO4 as an additive to the PVA-based gel electrolyte was reported as a perfect solution for lowering self-discharge, increasing energy, and lifetime of the EC [100]. Flexible capacitors based on gel electrolyte—poly (methyl methacrylate)-propylene carbonate-lithium perchlorate electrolyte with HQ as a neutral redox additive were also investigated [104].

## **4. Summary**

Redox-active electrolytes can be successfully applied in electrochemical capacitors and these electrolytes remarkably improve the energy density. It is crucial to use redox additives with a well-reversible and well-defined redox response, as the efficiency of charging/discharging should not be affected by redox process.

A variety of redox couples can be selected depending on the user's requirements: for the systems based on aqueous or nonaqueous electrolyte, with redox species

supposed to be active at the positively or negatively polarized electrode, or which parameters are the most important—high energy, high power, or very long cycle life. Taking into account aqueous-based redox-active electrolytes, the most attractive from the practical point of view are cationic and anionic electroactive species—because of their good solubility in water ensuring high conductivity of the solution. Moreover, the cells operating in organic/polymer-based electrolytes are more expensive due to the proton/ions permeable membranes that have to be used.

There are also a few issues that need to be solved. Redox species cause higher selfdischarge of the cell in comparison to ECs with pure EDL formation. Therefore, it would be beneficial to "trap" the redox species within the pores of the material to prevent their movement to the electrolyte bulk. Moreover, the lifetime of EC with redox-active electrolytes should be prolonged, because it is still significantly shorter than the lifetime of the cell operating in the typical capacitive electrolytes.

Nevertheless, redox-active electrolytes in electrochemical capacitors offer an interesting alternative to the solid-state compounds and composites with maintained power and improved charge/discharge efficiency.

## **Acknowledgements**

European Research Council Starting Grant 2017 project "IMMOCAP" (GA 759603) is acknowledged for financial support covering The Open Access Publishing Fee.

## **Author details**

Paulina Bujewska, Przemysław Galek, Elżbieta Frąckowiak and Krzysztof Fic\* Poznan University of Technology, Institute of Chemistry and Technical Electrochemistry, Poznan, Poland

\*Address all correspondence to: krzysztof.fic@put.poznan.pl

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Redox Mediated Electrolytes in Electrochemical Capacitors DOI: http://dx.doi.org/10.5772/intechopen.104961*

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## **Chapter 3**

## Redox Transitions in Pseudocapacitor Materials: Criteria and Ruling Factors

*Sergey N. Pronkin, Nina Yu. Shokina and Cuong Pham-Huu*

## **Abstract**

Pseudocapacitance is a phenomenon of charge storage involving redox transitions at the electrode/electrolyte interface. As the result of an electrode potential modulation, one or few components of the electrode and/or electrolyte change its/their oxidation states. The redox reaction may be confined to the interface or propagate into the bulk of the electrode material, thus significantly increasing the charge (and energy) capacitance of the material. The rate and the reversibility of the interfacial redox reaction are the key factors determining the efficiency of charge storage due to pseudocapacitance phenomena. The influence of the characteristics of the interfacial redox reaction on the efficiency of charge storage in pseudocapacitive materials is considered in the current chapter. In particular, the similarities and the differences between the charge storage in batteries and pseudocapacitors are discussed. The analysis of the pseudocapacitive behavior of electrode material using the impedance spectroscopy is presented.

**Keywords:** pseudocapacitance, ion insertion, interfacial kinetics, impedance, staircase model

## **1. Introduction**

The transition toward sustainable energy grids, based on renewable and noncritical resources, relies strongly on the technologies of efficient and reversible energy storage. Electrochemical energy storage devices (EESD) are among the most promising and versatile contemporary devices [1, 2]. Batteries, supercapacitors, and hybrid cells are the main classes of EESD, providing a wide range of energy and power densities. The energy densities of modern batteries are still inferior comparing to traditional (e.g., diesel) and modern (e.g., liquid hydrogen) fuels. However, high storage efficiency and zero operation emission make them indispensible elements of sustainable energy grids (**Table 1**).

Supercapacitors have ca. 10 times lower energy density comparing to batteries, but may operate at much higher power and have high capacitance retention with cycling. These differences are related to different mechanisms of charge storage in batteries and supercapacitors. Namely, in batteries the interfacial electrochemical reaction


### **Table 1.**

*Energy and power density of various energy storage devices.*

results in the transformation of the bulk of electrode materials. For example, in Li metal batteries, these reactions are the dissolution of Li to Li<sup>+</sup> at negative electrode and the intercalation of Li<sup>+</sup> to positive host electrode (e.g., graphite) [4]. While the maximal energy storage of batteries is primarily determined by the intercalation capacitance of host material, the power of energy input/output is limited by the rates of interfacial charge transfer and ion mass transport in solid-state bulk. Due to these factors, the energy capacitance of electrode materials in batteries strongly depends on the rate of charging.

On the other hand, in electrostatic capacitors, the charge *Q* is stored on two electrode plates separated by an insulator:

$$C = \frac{dQ}{dE} = \frac{\varepsilon\varepsilon\_0}{d}A,\tag{1}$$

where *<sup>ε</sup>*<sup>0</sup> <sup>¼</sup> <sup>8</sup>*:*<sup>854</sup> � <sup>10</sup>�<sup>12</sup> <sup>F</sup>*:*m�<sup>2</sup> is the vacuum permittivity, *<sup>ε</sup>* is the dielectric constant of the insulator, *d* is the distance between electrodes, and *A* is the surface area of the electrode.

Contrary to batteries, the capacitance of a capacitor does not depend on the rate of charging, providing that the charging time is longer than the circuit time constant *τ* ¼ *RESR* � *C*, where *RESR* is the equivalent serial resistance of the circuit (**Figure 1A**).

In electrochemical capacitors, the charge is stored at or in the vicinity of the electrode/electrolyte interface (**Figure 1B**). Comparing to electrostatic capacitors, electrochemical capacitors have much shorter *d* (order of nm) and higher specific surface area (SSA) *A* per mass unit. Thus, electrochemical capacitor possesses, in general, the charge capacitance ca. 4 orders of magnitude higher than electrostatic capacitors.

From the practical point of view, it is useful to define various types of specific capacitance, as listed below.


*Redox Transitions in Pseudocapacitor Materials: Criteria and Ruling Factors DOI: http://dx.doi.org/10.5772/intechopen.104084*

#### **Figure 1.**

*The scheme of distributions of the electrical potential between the electrodes of a solid-state capacitor (A) and an electrical double layer capacitor (B).*

In this case, *Cg* ¼ *Cs* � *SSA*. *Cg* is a useful parameter for comparison of different electrode materials, because *Cg* depends on both interfacial charge storage efficiency (characterized by *Cs*) and material structure (characterized by *SSA*). However, this comparison can be misleading due to the utilization of dry material mass with pores filled by air. The efficient electrode material has an open porous structure with pores filled by electrolyte. It results in a significant increase in the mass of the electrode material. Thus, the true value of *Cs* is much lower than the value calculated with the mass of dry material.

• Volume specific capacitance *CV* ¼ *C=V*. Similarly to *Cg* , in material science the value of *CV* is most often calculated using the volume unit of the electrode material *V*: *CV* ¼ *Cs* � *SSA* � *ρ*, where *ρ* is the density of a porous material. Contrary to *Cg*, *CV* has the same value for dry material and material filled with an electrolyte. This is one of the reasons why *CV* is a more relevant characteristics of the charge capacitance of the material than *Cg* [5].

Depending on the main mechanism of charge storage, one distinguishes between 2 types of supercapacitors: electrochemical double layer capacitors (EDLC) and pseudocapacitors.

As the name suggests, in EDLC main mechanism of charge storage is the formation of an electrical double layer at the electrode/electrolyte interface. In pseudocapacitors,

the predominant mechanism of charge storage is the pseudocapacitance phenomena. This phenomenon involves reversible interfacial electrochemical reaction and intercalation of a solvent component into the bulk of the electrode material. From the chemical point of view, the intercalation reactions in batteries and pseudocapacitors are similar. The important difference is that the rate of intercalation in supercapacitors, as opposed to batteries, is not limited by solid-state ion transport, but limited mostly by material intercalation capacitance and interfacial charge transfer rate.

In the current chapter, the influence of the rate of interfacial charge transfer on the electrochemical performance of electrode materials in supercapacitors is considered. The particular features of EDLC and pseudocapacitor performances are described. The focus of the current work is on the electrode materials of pseudocapacitors, because the charge transfer rate strongly influences the rate of pseudocapacitance phenomena. The influence of pseudocapacitance on the performance of EDLC is briefly discussed. The influence of interfacial charge transfer rate on the material capacitance of pseudocapacitors electrodes is first considered for a model flat electrode/electrolyte interface. Then, the dependence of the capacitance of porous electrode materials on interfacial charge transfer rate is analyzed. To evaluate the role of interfacial charge transfer, the models of interfacial impedance in the presence of pseudocapacitance reaction are considered.

## **2. The influence of the interfacial charge transfer rate on the performance of supercapacitors**

## **2.1 Electrical double-layer capacitors**

In EDLC the predominant mechanism of charge storage is charging of interfacial electrical double layer. This layer consists of adsorbed ions of electrolyte and solvent dipole molecules. The change of electrode potential results in changes of the ratio of adsorbed cations and anions at the interface and re-orientation of solvent dipoles. These phenomena result in charge accumulation at the interface with potential change, similarly to the charging of a capacitor when voltage is applied. The detailed models of the structure of the electrical double layer were developed in the last century [6]. As a very simple approximation, electrical double layer can be represented as a capacitor with one plate being an electrode and another plate being adsorbed ions and solvent molecules (**Figure 1B**). Under this approximation, the charge capacitance of the double layer *Cdl* is given by (1).

Considering that for aqueous solutions *ε* ¼ 80 and the thickness of electrical double layer is *d*≈1 nm, the surface specific charge capacitance *Cs* of electrode/ electrolyte interface can be estimated as *Cs* <sup>≈</sup><sup>2</sup> � <sup>10</sup>�<sup>5</sup> <sup>F</sup>*:*cm�<sup>2</sup> <sup>¼</sup> <sup>20</sup> *<sup>μ</sup>*F*:*cm�2. A small thickness *d* of electrical double layer allows using the materials with highly developed surface as EDLC electrodes, for example, highly porous carbon materials with *SSA* >2600 m<sup>2</sup>*:*g�<sup>1</sup> [7]. On the other hand, carbon/electrolyte interface has lower surface specific capacitance *Cs* comparing to metal/electrolyte interface, namely below 20 *μ*F*:*cm�<sup>2</sup> [8]. For a single-layer *sp*<sup>2</sup> carbon graphene sheet, the value of 13.5 *μ*F*:*cm�<sup>2</sup> is predicted and expected to decrease with the thickness of the carbon stack down to ca. 5 *μ*F*:*cm�<sup>2</sup> for the graphite structure. Thus, for carbon materials with highly developed SSA, the specific capacitance *Cg* related to double layer formation is expected to be in the range of 150–200 F/g.

## *Redox Transitions in Pseudocapacitor Materials: Criteria and Ruling Factors DOI: http://dx.doi.org/10.5772/intechopen.104084*

The modulation of electrode potential results in the change of ion adsorption, solvent dipoles orientation, and, possibly, electrode surface reconstruction [9]. These phenomena lead to the changes of the values of *d* and *ε* of (1). Thus, the value of specific interfacial capacitance *Cs* may depend strongly on electrode potential. This results in the a complex shape of CV curves with well-defined CV peaks and different values of double layer charging current at different potentials.

However, these interfacial phenomena are relatively fast and the total charge associated with them does not depend on the rate of charging (rate of potential sweep). Therefore, the value of charging current is expected to be proportional to the rate of potential sweep for a given potential. This proportionality is one of the criteria that was proposed to distinguish between the electrode materials behaving as supercapacitor or battery electrode materials [10].

The surface-specific charge capacitance can be increased due to the phenomena of pseudocapacitance, which involves a fast and reversible surface electrochemical transformation of an electrode component at a certain potential. The following electrochemical phenomena may occur: oxidation/reduction of surface oxides, partial charge transfer to/from adsorbed electrolyte species (for example, electrochemical adsorption/desorption of atomic hydrogen [11], underpotential deposition of electrolyte metal cations [12]).

For carbon electrodes, the characteristic surface redox transformation is attributed to the presence of quinone-type surface groups on the partially oxidized surface [13, 14]. This reversible surface transition is reflected by the appearance of a pair of CV peaks around the equilibrium potential *E*<sup>0</sup> ¼ 0*:*668 V (RHE) of quinonehydroquinone redox couple. The surface density of quinone groups on oxidized carbon surfaces is estimated to be approximately equal to 0*:*<sup>1</sup> � <sup>1</sup>*:*1%*at:* or 10�<sup>10</sup> � <sup>10</sup>�<sup>11</sup> mol*:*cm�<sup>2</sup> [14]. These groups provide additional 1 � <sup>10</sup> *<sup>μ</sup>*F*:*cm�<sup>2</sup> to surface-specific capacitance. The interfacial capacitance is also increased in the presence of heteroatoms in carbon structure. In particular, an increase in surface charge capacitance due to N-doping of carbon was studied most thoroughly [15–18]. N-doping of carbon materials up to few %*wt:* can be achieved, resulting in a significant increase in surface specific capacitance. It allows to reach the mass specific capacitance close or even higher than 300 F/g for N-doped carbon electrode materials [17, 18].

As mentioned above, the predominant mechanism of charge storage in EDLC is a charging of an interfacial electrical double layer by ion adsorption and solvent molecule orientation, even though the surface pseudocapacitance phenomena may provide additional contributions. In contrast, in pseudocapacitors, the contribution of pseudocapacitance is predominant due to the propagation of the surface pseudocapacitance reaction further to the bulk of electrode material.

## **2.2 Pseudocapacitors**

Even in the case of carbon materials, the pseudocapacitance reaction may propagate from interface further to the bulk of electrode material. One interesting example of this phenomenon is a hydrogen intercalation into a graphitic carbon structure. This phenomenon has been studied in details on activated carbon CH900–20 ACC (Japan) with SSA = 1520 m<sup>2</sup>*:*g�<sup>1</sup> [19, 20]. Few hours of galvanostatic cathodic polarization of CH900– 20 resulted in a formation of an intercalation compound with stoichiometry C6H:

$$\text{C}\_{\text{6(s)}} + \text{H}^{+}\_{\text{(l)}} + \text{e}^{-} \rightleftharpoons \text{C}\_{\text{6}}\text{H}\_{\text{(s)}} \tag{2}$$

The maximal charge capacitance provided by the reaction (2) is 943 F/g, and the total mass specific capacitance *Cg* of CH900–20 in acid electrolytes is equal to 1114.3 F/g. To the knowledge of the authors, this is the highest mass-specific capacitance reported for carbon materials. On the other hand, the estimated diffusion coefficient *<sup>D</sup>* of hydrogen in carbon, according to [20] is estimated as 5 � <sup>10</sup>�<sup>17</sup> cm2*:*s�1, and the intercalation is controlled by solid-state ion transport. This makes the material CH900–20 to be impractical for supercapacitors.

The pseudocapacitance is the main mechanism of charge storage in transition metal oxide (TMO) electrode materials commonly used as pseudocapacitor electrodes [21–23]. Similar to the functional groups at the carbon surface considered above, the metal cations at or in the vicinity of the electrode/electrolyte interface can undergo a fast reversible transformation due to a variability of oxidation states of transition metals. In comparison to the carbon, the percentage of metal sites available for the redox transformation is much higher: every metal atom is capable to change its oxidation state and participate in charge storage. This phenomenon provides significantly higher pseudocapacitance values for metal oxides comparing to carbon materials. The values above 900 F/g are routinely observed for RuO2-based electrode materials, which are among the best performing oxide materials in pseudocapacitors. As for the carbon electrodes, different phenomena can be involved in the pseudocapacitance: the surface redox transformation (involving interfacial charge transfer) and ion intercalation into solid-state electrode. For example, for RuO2, which is the most studied metal oxide supercapacitor material, the redox transition is described as follows [24]:

$$\mathrm{Ru}^{4+}\mathrm{O}\_{2} + \delta\mathrm{H}^{+} + \delta\mathrm{e}^{-} \rightleftharpoons \mathrm{H}\_{\delta}\mathrm{Ru}^{(4-\delta)+}\mathrm{O}\_{2} \tag{3}$$

The transformation starts as an electroreduction of RuO2 surface and then propagates into the bulk of the oxide. According to (3), this process can be also considered as H-intercalation. The transformation degree and the stored charge are characterized by the intercalation parameter *δ*: 0⩽*δ*⩽1.

In more general case, the reaction (3) can be presented as follows:

$$\rm{M\_xO\_y} + \delta \rm{Ct^+} + \delta \rm{e^-} \rightleftharpoons \rm{M\_xO\_yCl\_\delta} \tag{4}$$

The nature of intercalating cation into the oxide materials from aqueous electrolytes is still debated even for Mn oxides—the second most studied type of oxides for supercapacitor electrodes. Most of transition metal oxides utilized in supercapacitor electrodes are stable only in neutral and alkaline solutions. Thus, the intercalation of cations of electrolyte (K<sup>+</sup> , Na<sup>+</sup> ) was proposed to be a pseudocapacitance reaction (4) [25, 26]. The inclusion of Na and S elements into thin film of MnO2 and the decrease in Mn oxidation state from +4 to +3 (as the result of its polarization) have been indeed confirmed by ex situ XPS study [27]. However, the inclusion of these elements was mostly confined to the sub-surface atomic layers of oxide, while the high value of charge storage capacitance suggests the propagation of pseudocapacitance reaction into the bulk of oxide [27, 28]. Moreover, the pseudocapacitance reaction for Mn oxides appears to be pH-dependent: in general, higher charge capacitance is observed at higher pH [29, 30]. For Mn3O4/C electrode material, the potential of pseudocapacitance reaction (4) was found to shift by 59 mV/pH (**Figure 2**).

These facts are more consistent with the nature of pseudocapacitance reaction of Mn oxides in aqueous solutions as expressed by the following equation:

*Redox Transitions in Pseudocapacitor Materials: Criteria and Ruling Factors DOI: http://dx.doi.org/10.5772/intechopen.104084*

### **Figure 2.**

*CV curves of Mn3O4/C (*34%*wt:) oxide electrode in 1 M (Na2SO4 + NaOH) electrolytes with various pH, measured at 20 mV/s [30].*

$$\mathbf{M} \mathbf{n}^{4+} \mathbf{O}\_{\mathbf{x}, (\mathbf{s})} + \delta \mathbf{H}\_2 \mathbf{O} + \delta \mathbf{e}^- \rightleftharpoons \mathbf{M} \mathbf{n}^{(4-\delta)+} \mathbf{O}\_{\mathbf{x}-\delta} \mathbf{O} \mathbf{H}\_{\delta, (\mathbf{s})} + \delta \mathbf{O} \mathbf{H}\_{(\mathbf{l})}^- \tag{5}$$

The total specific amount of charge (in C/g) stored due to pseudocapacitance *Qpc* is as:

$$Q\_{pc} = \frac{\boldsymbol{\pi} \cdot \boldsymbol{\delta} \cdot \boldsymbol{F}}{M},\tag{6}$$

where *x* is the stoichiometric coefficient of metal cation in MxOy oxide, *M* is molecular weight of the oxide (in g/mol). For example, for RuO2, *Qcp* = 725.4 C/g, according to (6), which is close to the experimental values of specific charge density *QC* ¼ 614 C/g reported for amorphous RuO2 [24]. This indicates that *δ*≈0*:*85 for RuO2, i.e., the most of Ru cations in the bulk of the oxide are involved in the charge storage.

### **2.3 Pseudocapacitors versus batteries**

The propagation of reaction (4) into the bulk of the oxide results in the cation Ct<sup>+</sup> intercalation into the oxide MxOy. Cation intercalation is also a charge storage mechanism in various types of batteries: in particular, the Li<sup>+</sup> cations are intercalated into the graphite cathodes in the conventional Li batteries [3]. The apparent similarities between these intercalation processes sparkled the discussion about the criteria to distinguish between the materials behavior as battery or as supercapacitor electrodes [10, 31].

As discussed in Section 1, comparing to batteries, the pseudocapacitors provide smaller energy density storage, but operate with higher output power and better reversibility, i.e., retention of capacitance in a larger number of cycles. It suggests that the intercalation reaction is faster and more reversible in supercapacitors. The criteria of reversibility of intercalation reaction are considered below.

The rate of the reversible interfacial redox process of Ct+ intercalation (4) is expressed by the equation [32], similar to the Buttler-Volmer equation combined with the Frumkin isotherm:

$$i = i\_0 \left( (1 - \delta) e^{-(1 - a)\mathfrak{g}\delta} e^{\frac{(1 - a)\mathcal{F}(E - E\_0)}{RT}} - \delta e^{\mathfrak{g}\mathfrak{g}\delta} e^{-\frac{a\mathcal{F}(E - E\_0)}{RT}} \right). \tag{7}$$

Here *i*<sup>0</sup> is the exchange current density of reaction (4), *α* is the symmetry factor (*α* ¼ 0*:*5 for reversible intercalation). The constant *g* of the Frumkin isotherm characterizes the lateral interaction between the intercalated ions: *g* <0 for attractive and *g* >0 for repulsive interaction (most commonly observed for intercalation cations). For example, *<sup>g</sup>* ¼ �4*:*2 was found for intercalation of Li<sup>+</sup> into LixCoO2 [32].

The (7) assumes that the cations Ct+ in the electrolyte and in the solid oxide phase are in quasi-equilibrium, which depends only on the interfacial potential *E* according to the Frumkin isotherm:

$$\frac{\delta}{1-\delta} = e^{\frac{F(E-E\_0)}{kT}} e^{-g(\delta - 0.5)}.\tag{8}$$

Eqs. (7) and (8) assume that the distribution of *δ* within the bulk of host material is not influencing the interfacial reaction, i.e., that the rate of (4) is determined by its kinetics and not by the mass transport of Ct+ in the solid phase. For example, for Li intercalation into LiCoO2, this model is valid at a potential sweep rate 10–50 *μ*V*=*s. At faster charging rates, the rate of Li<sup>+</sup> intercalation is determined by its diffusion in the host material and the Eqs. (7) and (8) are no longer valid.

**Figure 3** illustrates the intercalation process of Ct+ into the oxide particle and the formation of a gradient of Ct+ concentration (or *δ*) within the particle. If *δ<sup>L</sup>* ≫ *d*, where *δ<sup>L</sup>* is a characteristic length of concentration gradient and *d* is the particle size,

**Figure 3.** *The intercalation of Ct<sup>+</sup> into the oxide particle.*

*Redox Transitions in Pseudocapacitor Materials: Criteria and Ruling Factors DOI: http://dx.doi.org/10.5772/intechopen.104084*

then the gradient of *δ* has no influence on the rate of interfacial reaction, the Eqs. (7) and (8) are valid, and the reaction (4) can be considered as reversible.

Assuming that the material of electrode has sufficiently high electronic conductivity, the presence of a gradient of potential inside the solid-state material can be neglected. Thus, the diffusion is a predominant mode of propagation of reaction (4) into the bulk of solid state. The solution of the first Fick's law for planar diffusion with boundary conditions *CCt*þ,*x*¼<sup>0</sup> <sup>¼</sup> *<sup>C</sup>*0,*CCt*þ,*x*¼<sup>∞</sup> <sup>¼</sup> 0 shows the relation between the current *<sup>I</sup>* of Ct+ intercalation reaction (4) and the length of the concentration gradient *δL*:

$$I = A \text{FC}\_0 \frac{D\_{\text{Cr}^+}}{\delta \text{L}}. \tag{9}$$

Here *A* is the surface area of the interface, *DCt*<sup>þ</sup> is the solid-state diffusion coefficient of Ct<sup>+</sup> . For an efficient intercalation host material, maximal value of *δ* is equal to 1 and *CCt*<sup>þ</sup> can be approximated as a concentration of metal cations at the interface, i.e. *CCt*<sup>þ</sup> <sup>¼</sup> *<sup>C</sup>*<sup>0</sup> � *<sup>δ</sup>*. For example, for MnO2, *<sup>C</sup>*<sup>0</sup> <sup>≈</sup> *<sup>ρ</sup>=<sup>M</sup>* <sup>¼</sup> <sup>0</sup>*:*058 mol*:*cm�3.

From the practial point of view, it is useful to relate the charging rate *I* (in A) with mass specific current density *im* (in A/g) by considering the geometry of a spherical particle with the diameter *d*:

$$i\_m = \frac{\text{€}IA}{\rho d},\tag{10}$$

where *ρ* is particles density (in g*:*cm�3). Combining (9) and (10), the equation for the dependence of *δ<sup>L</sup>* on *im* is obtained:

$$
\delta\_L = \frac{\mathsf{GFC}\_0 \delta D\_{\Omega^+}}{i\_m \rho d}. \tag{11}
$$

Eq. (11) provides the criterion of reversibility of reaction (4). If *δ<sup>L</sup>* ≫ *d*, then the concentration of intercalated cation within the oxide is nearly constant. In this case, the value of intercalation factor *δ* depends only on interfacial conditions, namely potential *E*, and the Eqs. (7) and (8) are valid. On the other hand, if *δ<sup>L</sup>* ≈*d*, then the value of intercalation factor *δ* depends on the distance from interface and is changing with time at constant *E*.

The strongest incertitude in Eq. (11) is related to the values of solid-state diffusion coefficient *DCt*<sup>þ</sup> at ambient temperature. For Li batteries electrode materials, the values of *DLi*<sup>þ</sup> are reported mostly for non-aqueous electrolytes. The values as high as *DLi*þ*=<sup>C</sup>* ¼ <sup>10</sup>�<sup>9</sup> � <sup>10</sup>�<sup>10</sup> cm<sup>2</sup>*:*s�<sup>1</sup> were determined for Li+ intercalation into graphite electrode using impedance spectroscopy [33]. However, the values of Li+ solid-state diffusion coefficients are most commonly found in the range 10�<sup>13</sup> � <sup>10</sup>�<sup>15</sup> cm<sup>2</sup>*:*s�<sup>1</sup> using the modeling of galvanostatic charging curves [34–36]. The advantages and limitations of this approach are thoroughly discussed in [37]. The thorough review of ambient temperature ionic conductivity in Li batteries electrode material is given in [38].

The diffusion coefficients *DCt*<sup>þ</sup> for cation intercalations from aqueous electrolytes are seldom measured. Indirectly, the values of *DCt*<sup>þ</sup> can be estimated from the ionic conductivity values of oxide *σ<sup>i</sup>* using the following Equation [38]:

$$
\sigma\_i = \frac{F^2 C\_{\Omega^+} D\_{\Omega^+}}{RT}.\tag{12}
$$

For example, for MnO2 oxides, *σ<sup>i</sup>* was reported in the range 0.001–0.02 Ohm�<sup>1</sup> *:*cm�1, depending on their crystallographic structure [39]. Thus, the diffusion coefficient values can be estimated as *DCt*<sup>þ</sup> <sup>≈</sup>10�<sup>7</sup> � <sup>10</sup>�<sup>9</sup> cm2*:*s�<sup>1</sup> for a cation insertion into MnO2 from aqueous electrolytes. In general, the rate of diffusion in supercapacitor electrode materials is expected to be few orders of magnitude larger comparing to Li<sup>+</sup> diffusion in Li batteries electrode materials. Using the Eqs. (11) and (12), one may estimate *δ<sup>L</sup>* value for 100 nm particles of a material with *σ<sup>i</sup>* ¼ 0*:*001 Ohm�<sup>1</sup> *:*cm�1: *<sup>δ</sup><sup>L</sup>* <sup>≈</sup> <sup>0</sup>*:*03 cm even at fast charging rates (*im* <sup>¼</sup> 100 A/g). Thus, for the oxide materials with relatively high ionic conductivity, *δ<sup>L</sup>* ≫ *d* even for fast charging rates. It indicates that the intercalation reaction can be considered as reversible and the material acts as a pseudocapacitor electrode.

The equations above provide qualitative criteria to distinguish between the materials behaving as supercapacitor or battery electrodes. To develop quantitative criteria, the modeling of various simultaneous interfacial processes as functions of their rates is needed. The simplest strategies for these models are considered below.

## **2.4 Impedance of flat interface in the presence of pseudocapacitance reaction**

Impedance spectroscopy may provides detailed information of the interfacial phenomena occurring with various rates. In particular, in the presence of pseudocapacitance reaction on the flat electrode/electrolyte interface the following processes have to be taken into account.

• Formation of electrical double layer by reversible adsorption of electrolyte ions. Assuming a small amplitude of potential modulation, the interfacial characteristics (dielectric constant *ε*, change of ion adsorption with potential *d*Γ*=dE*) can be considered to be constant. The impedance of double layer formation is then expressed as a capacitance impedance:

$$Z\_{dl} = -\frac{i}{a\mathcal{C}\_{dl}},$$

where *ω* ¼ 2*πf* is the angular frequency (in rad/s), *f* is the potential modulation frequency (in Hz), *Cdl* is the double layer capacitance, *Cdl* ¼ *Cs* � *A*, where *Cs* is the surface specific capacitance (in F*:*cm�2), *A* is the electrode area (in cm2). Depending on the electrode material, *Cs* varies between 5 *μ*F*:*cm�<sup>2</sup> for carbon electrodes and 20–50 *μ*F*:*cm�<sup>2</sup> for metal electrodes.

• Interfacial charge transfer in the course of the reaction (4). For small potential modulations around the equilibrium redox potential *E*0, the Eq. (7) can be simplified to a linear form. The current *i* is proportional to the overvoltage *E* � *E*<sup>0</sup> and the impedance of this process can be approximated by a simple resistance *Rct*:

$$Z\_{ct} = R\_{ct} = \frac{RT}{F \cdot i\_0}.$$

• Diffusion of charged species into the bulk of the oxide material. Assuming the semi-infinite diffusion conditions (*CCt*þ,*x*¼<sup>0</sup> ¼ *C*0,*CCt*þ,*x*¼<sup>∞</sup> ¼ 0), the impedance of this process is given by the warburg impedance:

*Redox Transitions in Pseudocapacitor Materials: Criteria and Ruling Factors DOI: http://dx.doi.org/10.5772/intechopen.104084*

$$\begin{aligned} Z\_W &= \frac{\sigma}{\sqrt{\alpha}} - \frac{i\sigma}{\sqrt{\alpha}}, \\ \sigma &= \frac{RT}{\sqrt{2D}AF^2C\_0}. \end{aligned}$$

• Accumulation of charge in the bulk of oxide material. For small potential modulation this process can be considered similar to the accumulation of the charge at a capacitor:

$$\begin{aligned} Z\_P &= -\frac{i}{a\mathcal{C}\_p}, \\ \mathcal{C}\_p &= \frac{\mathcal{C}\_\mathcal{g} \cdot A}{\mathcal{S}\mathcal{S}A \cdot 10^4}. \end{aligned}$$

Here *Cp* is the capacitance of charge accumulation due to the pseudocapacitance reaction (in F), *Cg* is the gravimetric capacitance of electrode material (in F/g), *SSA* is the specific surface area of electrode material (in m<sup>2</sup>*:*g�1).

For a small potential modulation, the listed processes can be considered to be independent. In this case, the total interfacial impedance can be expressed by the Frumkin-Melik-Gaykazyan (FMG) model [32, 33], depicted as an equivalent circuit in **Figure 4**.

From the practical point of view, it is useful to correlate the frequency of the potential modulation *f* in the impedance spectra with the typical rates of potential variation during the charge/discharge of supercapacitors. The correlation can be done by using the root mean square rate *vrms* of potential in the course of sinusoidal modulation:

$$v\_{rms} = \frac{2\pi E\_0 f}{\sqrt{2}},\tag{13}$$

where *E*<sup>0</sup> is the amplitude of potential modulation. One can demonstrate that 1C charging rate of a capacitor with 1 V operation voltage is comparable with the *vrms* at *f* ¼ 6*:*25 Hz (for *E*<sup>0</sup> ¼ 10 mV). For 10C charging rate, the equivalent frequency is 62.5 Hz. Thus, one can roughly define the frequency range of interest for supercapacitors as ca. 0.1–10 Hz, and for batteries—below 1 Hz. This attribution is important for an interpretation of impedance modeling results.

**Figure 4** shows the results of the modeling of interfacial impedance of a flat electrode in the presence of pseudocapacitance reaction, assuming *Cs* <sup>¼</sup> <sup>5</sup> *<sup>μ</sup>*F*:*cm�2, *Cg* <sup>¼</sup> 90 F*:*g�1, *SSA* <sup>¼</sup> 30 m<sup>2</sup>*:*g�1. The maximal surface specific capacitance of this interface is *Cm* <sup>¼</sup> <sup>305</sup> *<sup>μ</sup>*F*:*cm�2, consisting of both *Cdl* and *Cp* contributions. The Bode plots (**Figure 4B, D**) show the calculated values of the surface specific interfacial capacitance *Cs*,*<sup>c</sup>* as a function of the modulation frequency and the rates of interfacial charge transfer *i*<sup>0</sup> and solid-state diffusion *D*. The values of *Cs*,*<sup>c</sup>* are calculated from the calculated values of the interfacial impedance *Z*:

$$\mathcal{C}\_{\mathfrak{s}\mathfrak{c}} = -\frac{1}{\omega \cdot \mathrm{Im}(\mathbf{Z}) \cdot A} \,\mathrm{.}\tag{14}$$

As discussed above, the diffusion rate in supercapacitor electrode materials is few orders of magnitude larger comparing to Li batteries electrode materials. The

#### **Figure 4.**

*Nyquist (A,C) and bode (B,D) plots for the Frumkin-Melik-Gaykazuan model of interacial impedance in the presence of pseudocapacitance reaction, calculated for solid-state diffusion coefficients D* <sup>¼</sup> <sup>10</sup>�<sup>7</sup> cm<sup>2</sup>*:*s�<sup>1</sup> *(A,B) and D* <sup>¼</sup> <sup>10</sup>�<sup>15</sup> cm<sup>2</sup>*:*s�<sup>1</sup> *(C,D) and for various charge transfer rates (depicted in legend).*

influence of the rate of solid-state diffusion on the behavior of the electrode materials can be illustrated by the comparison of the Bode plots calculated by the FMG model with *<sup>D</sup>* <sup>¼</sup> <sup>10</sup>�<sup>7</sup> cm2*:*s�<sup>1</sup> for supercapacitors and with *<sup>D</sup>* <sup>¼</sup> <sup>10</sup>�<sup>15</sup> cm2*:*s�<sup>1</sup> for batteries electrodes.

In the first case (*<sup>D</sup>* <sup>¼</sup> <sup>10</sup>�<sup>7</sup> cm<sup>2</sup>*:*s�1), for not particularly slow charge transfer rate (*i*<sup>0</sup> > 10�<sup>4</sup> mA*:*cm�2) *Cs* is frequency-independent and close to *Cm*. This behavior is observed in relatively wide frequency range at low and moderate frequencies (*f* <100 Hz). This shape of the Bode plots shows that the electrode material, including its bulk, acts as capacitor, while neither rate of charge transfer, nor rate of diffusion, are limiting factors of its charge/discharge. This is the set of conditions under which the material acts as a pseudocapacitor. Under these conditions, the amount *δ* of intercalated ions is determined by the interfacial equilibrium according to Frumkin isotherm (8), and depends only on the electrode potential. The phase angle of the interfacial impedance is close to *<sup>ϕ</sup>* <sup>¼</sup> <sup>90</sup><sup>∘</sup> (dotted curves in **Figure 4B,D**). The Nyquist impedance plots under these conditions show straight vertical line with possible slight inclination due to the constant phase element behavior. The latter is often observed experimentally and explained by the capacitance slight dependence on applied frequency due to the surface heterogeneity, both chemical and geometrical [40, 41].

*Redox Transitions in Pseudocapacitor Materials: Criteria and Ruling Factors DOI: http://dx.doi.org/10.5772/intechopen.104084*

For very slow charge transfer rate (*i*<sup>0</sup> ≤10�<sup>5</sup> mA*:*cm�2), a clear decrease in *Cs* with an increase in *f* is expected even at low frequencies above 0.01 Hz, indicating that the material is not behaving as a pseudocapacitor even despite fast ionic transport in the solid-state.

At slow diffusion in solid state (*<sup>D</sup>* <sup>¼</sup> <sup>10</sup>�<sup>15</sup> cm2*:*s�1), the surface specific charge capacitance *Cs* decreases with increase in *f* even for fast charge transfer rates. In fact, the model curves for exchange rates *<sup>i</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> � <sup>0</sup>*:*01 mA*:*cm�<sup>2</sup> are nearly identical, demonstrating that the rate of charging process is determined by a slow diffusion rate. For lower *i*<sup>0</sup> the decrease in *Cs* with increase in *f* becomes much sharper, as both diffusion and interfacial charge rates become limiting factors. As the amount *δ* of intercalated ions is limited by these factors, it is no longer in agreement with the Frumkin isotherm and the material cannot be considered as pseudocapacitor electrode material.

The described simple model is valid for flat interface only, because it neglects the effect of the geometry of electrode surface and material deposit. However, in EESD, such as batteries and supercapacitors, porous materials with well-developed interfacial surface area are utilized. To model the impedance of electrodes of supercapacitors, the influence of porous structure has to be taken into account.

### **2.5 Impedance of porous electrodes for supercapacitors**

The interfacial impedance inside a pore is often modeled by the staircase-type equivalent circuit.

It was demonstrated that for the infinitely long cylindrical pore the interfacial impedance *Zp* is described by the DeLevi Eq. (15) [42]:

$$Z\_p = \sqrt{R|Z|}e^{\frac{i\phi}{2}},\tag{15}$$

where *R* is the electrolyte resistance within the pore, *Z* is the impedance of a pore wall/electrolyte interface, and *ϕ* is the phase angle of *Z*. The Eq. (15) predicts that phase angle of the interfacial impedance of the pore is equal to a half of the phase angle of flat interface with the same characteristics. For example, the phase angle of the impedance of the interface with a double layer formation only is 90<sup>∘</sup> for a flat interface, and 45<sup>∘</sup> for the interface inside the infinite pore.

The behavior of the interfacial impedance of a porous system is determined by the penetration depth *λ* of potential modulation signal into the pore [43]:

$$
\lambda = \frac{1}{2} \sqrt{\frac{\sigma d\_p}{2\rho \mathbf{C}\_s}},
\tag{16}
$$

here *σ* is the conductivity of the electrolyte, *dp* is the pore diameter. The Eq. (16) suggests that at the frequency below a certain threshold *f* <sup>0</sup>, *λ*>*lp*, where *lp* is the length of the pore, and the whole surface of pore acts as a flat interface. Thus, the transition from infinite pore behavior to flat interface is observed on the impedance curves, that is often detected in the experiments with porous electrodes (for example [44]).

This staircase model allows to correctly fit experimental impedance spectra of porous electrodes in the absence of interfacial faradaic processes and determine the capacitance *Cs*. Moreover, the average pore length *lp* can be estimated from the *f* <sup>0</sup>

value, providing that the average diameter *dp* of the pores is known from the other methods [45]. The model can also be adapted to take into account irregular shape of the pores: conical, globular pores etc. [43].

The majority of the studies of the impedance of porous systems simulate the pores as a system of parallel independent pores. The staircase circuit is applied separately to each pore, and total impedance is calculated as a parallel combinations of the pores. The supercapacitors electrode materials, for example the activated carbon, often have branching hierarchical structures: the larger pores are branching into few narrower pores of next generation, which can be branching further. The hierarchical porous structure of these materials makes the application of the staircase model more complicated, because the staircase circuits of pores from different generations are no longer independent.

The model of branching pores was proposed in [46] by introducing 2 generations of pores: the large (*μ*m size) voids between carbon particles and the narrower (sub-*μ*m size) internal pores. The larger pores were branching into *n* smaller pores with length and diameter scaled by *al* and *ad* factors comparing to the parent pore. This model clearly demonstrated the influence of various parameters of geometry of pores on the performance of carbon materials in supercapacitors. An important conclusion of this work was the role of the interplay between the different parameters of the geometry. In particular, as the porosity of the material was a fixed parameter in the model, an increase in branching factor resulted in an increase in the diameter scaling factor, i.e., narrower pores of the next generation. As it follows from (16), the AC-penetration depth *λ* is shorter for an narrower pore, and, therefore, the utilization of an interface within such pore is less efficient. Thus, it was concluded that the branching of pores is counter-productive for their efficient utilization [46].

The staircase model was recently generalized take into account complexity of the porous structure of carbon electrode materials of supercapacitors [47]. Three generations of pores were considered: short and wide pores (*d*<sup>1</sup> <sup>¼</sup> <sup>10</sup> � 30 nm) of the 1*st* generation were branching into *β*<sup>12</sup> narrow mesopores of the 2*nd* generation (*d*<sup>2</sup> <sup>¼</sup> <sup>3</sup> � 10 nm). The pores of the 2*nd* generation formed the main population of pores. These pores could also branch into *β*<sup>23</sup> micropores (*d*<1 nm). The branching was allowed to occur along the whole length of a parent pore. To take into account the contribution of pseudocapacitance, the *Z <sup>f</sup>* interfacial impedance (see **Figure 5**) was

#### **Figure 5.**

*The staircase equivalent circuit of the impedance model inside a single pore. The highlighted elements form the repeating pattern along the pore.*

*Redox Transitions in Pseudocapacitor Materials: Criteria and Ruling Factors DOI: http://dx.doi.org/10.5772/intechopen.104084*

modeled as a serial combination of charge transfer resistance *Rct* and pseudocapacitance *Cp*. To limit the number of model parameters, relatively fast ion transport in the solid phase of the electrode was assumed. Thus, the warburg impedance *ZW* was excluded from this model.

The Nyquist plots of the interfacial impedance, modeled by generalized staircase model in the absence of the faradaic impedance, are shown in **Figure 6A**. In agreement with simpler models, the curves show the transition from infinite pore behavior (*<sup>ϕ</sup>* <sup>¼</sup> <sup>45</sup><sup>∘</sup> ) at high frequency to flat capacitor behavior (*<sup>ϕ</sup>* <sup>¼</sup> <sup>90</sup><sup>∘</sup> ) at frequencies *f* > *f* <sup>0</sup>. In the presence of the fast pseudocapacitance reaction (*i*<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*1 mA*=*cm2) and high pseudocapacitance (*Cp* <sup>¼</sup> <sup>100</sup> *<sup>μ</sup>*F*:*cm�2), a characteristic semi-circular behavior is observed at intermediate frequencies (**Figure 6A,C**). The model also shows that the interfacial impedance changes significantly with the geometry of the electrode material for the same characteristics of pseudocapacitance reaction (**Figure 6A,C**).

One may distinguish between two main factors of the influence of the geometry of electrodes on the performance of electrode materials in supercapacitors:" penetration depth" (*λ*-factor) and" porosity" (*ρ*-factor). The *λ*-factor is related to the decrease in the value of *λ* with an increase in potential modulation frequency *f* (16). As the

#### **Figure 6.**

*The Nyquist (A,C) and bode (B,D) plots for porous electrodes calculated by the generalized staircase model with 3 generations of pores. The non-faradaic model excludes Z <sup>f</sup> from the calculations.*

surface of" active" interface within the pores decreases, the interfacial capacitance *Cs* is also decreasing with increase in *f*, and this decrease is more pronounced for smaller and longer pores. On the other hand, smaller pores have lower inner volume and can provide high *SSA* while having lower porosity and higher density *ρ* of the material. Thus, these two factors are often found to be counter-active.

As it was mentioned above, the utilization of volume specific capacitance *CV* ¼ *Cs* � *SSA* � *ρ* allows accounting for both *λ*- and *ρ*- factors. Similar to the previous section 2.4, *Cs* can be calculated from the model impedance values by (14). Similar to the model of flat interface (**Figure 4B,D**), the Bode plots for porous electrodes, constructed with *CV* (**Figure 6B,D**), demonstrate flat segments at low frequencies, corresponding to *<sup>ϕ</sup>* <sup>¼</sup> <sup>90</sup><sup>∘</sup> , or pseudocapacitive behavior. The values of *CV* of these segments are determined by *ρ*-factor: it is higher for smaller and longer pores. Also, the branching of the pores provides the material with higher density (fewer pore are needed for high *SSA*), and, thus, higher *CV* values at low frequencies (**Figure 6B**).

However, comparing to flat interface, the segment of pseudocapacitive behavior is much shorter: the decrease in *CV* with increasing frequency is observed at significantly lower frequencies. This decrease is related to *λ*-factor of geometry influence and it is more significant and observed at lower threshold *f* <sup>0</sup> for smaller and longer pores (**Figure 6D**).

For fast pseudocapacitance reaction (*i*<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*1 mA*:*cm�2) and the typical porous structure of mesoporous carbon (predominant pores with *d*≈ 3 � 10 nm and *l*≈ 1 � 3 *μm*), the pseudocapacitive behavior is observed up to *f* ≈100 Hz, i.e., in the region of potential modulation rates relevant for supercapacitors. However, for slower reactions the range of pseudocapacitive behavior is limited to low-frequency ranges only (*f* <1*Hz*), showing strong effect of *λ*-factor of the geometry of pores on the capacitance of electrode material. Also, for longer pores of few tens of *μm* (e.g., thick electrode deposits), the pseudocapacitive behavior is restricted to slow potential modulation even for very fast pseudocapacitance reaction (**Figure 6D**).

## **3. Conclusions**

The phenomenon of pseudocapacitance allows to increase significantly the charge storage capacitance of the materials by involving the bulk of the electrode material to the charge storage. This phenomenon involves a fast and reversible interfacial charge transfer reaction, followed by an insertion of electrolyte cation into sub-surface and bulk layers of the solid-state electrode. This phenomenon is similar to cation intercalation into the batteries electrodes. However, in the case of supercapacitors, the interfacial reaction must be fast and reversible. In this case, the amount of inserted cation, as characterized by the intercalation factor *δ*, is in agreement with the Frumkin isotherm (8). In practical terms, this criterion means that the amount of stored charge depends only on the applied potential and does not depend on the rate of charging.

In the case of several simultaneous interfacial phenomena, the electrochemical behavior of the materials can be analyzed by interfacial impedance measurements and modeling. The advantage of the impedance spectroscopy is that, due to a small potential modulation, various interfacial phenomena can be considered to be independent and total impedance can be modeled as a combination of several elements corresponding to different processes. Namely, the impedance of flat electrode/electrolyte interface in the presence of pseudocapacitance phenomena can be analyzed by the FMG model. This analysis shows that in the case of relatively fast ion transport in

the solid phase (*D* ≈10�<sup>7</sup> cm2*:*s�<sup>1</sup> ) and not particularly slow interfacial charge transfer rates (i.e., *i*<sup>0</sup> > 10�<sup>4</sup> mA*:*cm�2), the pseudocapacitive behavior is expected up to very fast potential modulation (*f* ≈100 Hz). Under these conditions, the calculated surface specific interfacial capacitance *Cs* does not depend on the frequency *f* and the impedance phase angle is close to *<sup>ϕ</sup>* <sup>¼</sup> <sup>90</sup><sup>∘</sup> . This behavior is in clear contrast to the behavior of materials with slow ion transport in solid phase *D* ≈10�<sup>15</sup> cm2*:*s�1, in which case a decrease in *Cs* with increase in potential modulation rate is detected already at low frequencies *<sup>f</sup>* <sup>≈</sup>0*:*01 Hz even for fast charge transfer rates (*i*<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*1 mA*:*cm�2).

The analysis of the electrochemical behavior of electrode materials for supercapacitors must take into account the complex porous structure of these materials. For this purpose, the generalized staircase model of interfacial impedance for materials with hierarchical branching porous structure can be used. The effects of the geometry of pores on the performance of electrode materials of supercapacitors can be roughly categorized into two groups of factors. First, the presence of pores allows increasing significantly the specific surface area of the material, thus increasing the area of interface available for charge storage. The narrower pores with higher branching factors have lower inner volume, resulting in higher density of electrode material and, correspondingly, higher volume specific capacitance *CV*. This group of effects is referred to as *ρ*-factors. The second group of effects, referred to as *λ*-factors, is related to the dependence of *λ*, i.e., penetration depth of potential modulation to the pore, on the potential modulation frequency *f*. The value of *λ* is shorter for narrower pores (16); thus *ρ*- and *λ*-factors are most often counter-active.

The modeling of the interfacial impedance of porous electrodes shows that the pseudocapacitive behavior (i.e., the values of *Cs* and *CV* being independent on the potential modulation rate *f* ) is confined to lower frequencies *f* due to *λ*-factors, comparing to flat interface. In particular, for thick deposits of mesoporous carbon, a decrease in interfacial capacitance with increase in *f* can be expected even for relatively fast interfacial charge transfer rate *i*<sup>0</sup> ≈0*:*01 mA*:*cm�2.

In general, one may conclude that in the case of flat electrodes of materials with sufficiently high ionic conductivity (*σ<sup>i</sup>* >0*:*001 Ohm�<sup>1</sup> *:*cm�1), the pseudocapacitive behavior of electrode materials can be observed even for moderate interfacial charge rates (*i*0⩾10�<sup>4</sup> mA*:*cm�2). However, the materials with high *SSA* and developed porous structure are commonly employed in supercapacitors, which imposes stronger limitations on the rate of interfacial charge transfer. Thus, the demonstration of the pseudocapacitive behavior of a material with flat geometry of the interface is not sufficient to suggest its application in supercapacitors. The pseudocapacitive behavior has to be experimentally demonstrated and/or numerically simulated for the given material with porous structure.

## **Acknowledgements**

Sergey Pronkin is indebted to ANR for the financial support of the project INFINE (ANR-21-CE08-0025, 2021-2024).

## **Conflict of interest**

Authors declare no conflict of interests.

## **Abbreviations**


## **Author details**

Sergey N. Pronkin<sup>1</sup> \*, Nina Yu. Shokina<sup>2</sup> and Cuong Pham-Huu<sup>1</sup>

1 Institute of Chemistry and Processes for Energy, Environment, and Health (ICPEES UMR-7515 CNRS-Unistra), Strasbourg, France

2 University of Freiburg, Medical Center, Radiology Clinics, Freiburg, Germany

\*Address all correspondence to: sergey.pronkin@unistra.fr

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Redox Transitions in Pseudocapacitor Materials: Criteria and Ruling Factors DOI: http://dx.doi.org/10.5772/intechopen.104084*

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Section 3
