Other than self-discharge, our discussion of EDLCs has not involved faradaic reactions; that is, we have assumed that no charge is transferred across the double layer, and that there is no change in oxidation state due to reaction. The resulting current–voltage behavior is purely capacitive and would approximate the ideal box shown earlier in Figure 11.9.
The behavior of real systems can vary somewhat from this. For example, the current potential diagram for a carbon black electrode subjected to a CV is shown in Figure 11.21. Notice that the basic shape is somewhat rectangular, as we would expect for an EDLC. Additionally, there are noticeable peaks around 0.55 V. These are reversible reactions on the carbon surface, attributed to the hydroquinone/quinone (HQ/Q) reaction. During the positive scan, HQ on the carbon surface is oxidized to Q, but the reaction does not continue indefinitely. Once the HQ on the surface is completely oxidized, the reaction stops. The reaction is reversed on the negative sweep, converting Q back to HQ. Although the CV looks more or less like that of a capacitor, some of the current is due to charging of the electrical double layer and some is due to faradaic reaction of the HQ/Q redox couple. Is this a battery or capacitor? The answer is yes, it has characteristics of both a battery and a capacitor. We see that there isn’t always a clear distinction in the behavior. However, for typical carbon EDLCs, no more than 5% of the current is due to the reactivity of surface oxides. Thus, these carbon materials are still treated as electrochemical double-layer capacitors.
Now imagine an electrode also undergoing a faradaic reaction, but where the potential changes linearly with charge. Physically, this behavior could occur from a redox reaction that is limited to the electrode surface, or from adsorption. Recalling Equation 11.2, the change in potential with charge defines a differential capacitance. Here, the physical origin of the capacitance does not arise from charge accumulation in the double layer; rather, it is the result of a faradaic reaction. We will use a new symbol, Cϕ, to represent this so-called pseudo-capacitance. A device such as the one we have described would also have a capacitance associated with double-layer charging, but here 
. A device with both double layer and pseudo-capacitance may be represented by the equivalent circuit shown in Figure 11.22. Here these capacitances are in parallel, and the double-layer capacitance is typically much smaller than the pseudo-capacitance. A characteristic feature of pseudo-capacitors is that the faradaic reaction is more or less restricted to the surface. Thus, the charge-transfer process is limited by the surface area available.
A common example of pseudo-capacitance is ruthenium oxide. During charging and discharging,
(11.48)
Equation 11.48 is a faradaic reaction and involves the transfer of electrons to and from the metal oxide; however, because the reactions are limited to the surface, achieving a high surface area is critical for high energy storage. A typical CV for a pseudo capacitor is shown in Figure 11.22. The ideal rectangle is approximated and the capacitance is taken as
Capacitance values for good pseudo-capacitor materials can be 5–10 F·m−2 of actual surface area, compared to typical CDL values of approximately 0.25 F·m−2. From Figure 11.22, the average current density (based on the superficial area) is ∼25 A·m−2. The data in Figure 11.22 correspond to a capacitance per superficial or geometric electrode area of about 420 F·m−2. Typically, pseudo-capacitors have a greater capacitance than EDLCs, but tend to be much more expensive due to higher materials costs.
EDLCS AND PSEUDO-CAPACITORS
EDLCs store energy electrostatically in the electrical double layer through charge separation; no electron-transfer reactions are needed for EDLC energy storage. In contrast, with pseudo-capacitors, a highly reversible faradaic reaction occurs on the surface, and the energy is stored chemically as a result of electron transfer. Because the charge passed depends linearly on the potential, the behavior of these materials mimics that of a capacitor. Since there are faradaic reactions, they are called pseudo-capacitors.
Closure
In this chapter, we introduced two types of electrochemical capacitors: EDLCs and pseudo-capacitors. Electrochemical double-layer capacitors do not rely on faradaic reactions for energy storage. Porous electrodes with high-surface areas are a key feature for these energy storage devices. The analysis of their behavior was examined using the same electrochemical principles that we applied to batteries and fuel cells. Notably, these devices are characterized by high-power density and relatively low-energy density. Pseudo-capacitors exhibit performance characteristics similar to EDLCs, but faradaic reactions are present.
Further Reading
- Bequin, F. and Frackowiak, E. (2013) Supercapacitors: Materials, Systems and Applications, John Wiley & Sons, Inc.,
- Conway, B.E. (1999) Electrochemical Supercapacitors: Scientific Fundamentals and Technological Applications, Springer.
- O’M Bockris, J. and Reddy, A. K. N. (1970) Modern Electrochemistry, Vol. 2, Plenum Press, New York.
- You, A., Chabot, V., and Zhang, J. (2013) Electrochemical Supercapacitors for Energy Storage and Delivery: Fundamentals and Applications, CRC Press.
Problems
11.1. In Illustration 11.1, the specific capacitance of carbon was calculated to be 150 F·g−1. To fabricate an electrochemical double-layer capacitor, even if the separator, current collector, and packaging weights are ignored, the theoretical value for capacitance in F·g−1 must be reduced by a factor of exactly 4. Why?
11.2. Derive Equation 11.8 for a 1 : 1 electrolyte. Hint: Start with Poisson’s Equation 2.32 and follow the development in Section 2.13. Hint: Use Cartesian coordinates and do not make the assumption of a small potential. Finally, the following transform is helpful:
11.3. Electrolytic capacitors have a polarity and will be destroyed if subjected to even a modest voltage of more than about 1.5 V of the wrong polarity. Explain this behavior. What would be the effect of connecting two electrolytic capacitors with polarity in series?
11.4. Sketch the charge density and potential across a double layer that includes both charge in the compact layer near the electrode (OHP) and the diffuse layer (GC). Assume the metal is positively charged. Develop Equation 11.9 showing that the two capacitances combine in series.
11.5. Derive Equation 11.11 using the geometry of Figure 11.6. Assume that the region of low dielectric constant includes the first row of water on the electrode and a second region of high dielectric constant that extents from the ion at the OHP to the first row of water (2rw).
11.6. A carbon porous electrode EDLC uses 0.05 M KOH at a temperature of 25 °C for the electrolyte. At the point of zero charge (PZC) and at a potential 0.1 V from the PZC, calculate the following capacitances per unit area [F·m−2]:
- Helmholtz contribution to the double-layer capacitance. Rather than the bulk dielectric constant of water, use a value of 11 for this inner region. The atomic diameter of K is 0.46 nm.
- Contribution of the diffuse layer from the theory of Gouy and Chapman.
- Combined value for the capacitance.
- Typical experimental values for the double-layer capacitance in aqueous solution are 0.2–0.5 F·m−2. Discuss importance of using the corrected dielectric constant for the Helmholtz region.
11.7. The table includes some data from Bockris and Reddy (Table 7.11) for 0.1 N aqueous chloride solutions. (a) for a 0.1 N solution, show that COHP has a relatively small effect on the differential capacitance. (b) What can be inferred about the adsorption of ions? (c) Using 0.14 nm for the radius of water, ɛrH = 40, and ɛrL = 6, does Equation 11.11 fit the data? Assume these are taken at room temperature.
| Ion | ri, Unsolvatedradius [nm] | Measuredcapacitance [F·m−2] |
| Li+ | 0.060 | 0.162 |
| K+ | 0.133 | 0.170 |
| Mg2+ | 0.065 | 0.165 |
| Al3+ | 0.050 | 0.165 |
| La3+ | 0.115 | 0.171 |
11.8. Describe how the structure of electrode might be designed differently for aqueous and nonaqueous electrolytes.
11.9. If the space between two parallel oppositely charged, infinite plates is comprised of two regions of different permittivity, how is the capacitance expressed?
11.10. The differential capacitance of a single electrode [F·m−2] is fitted with the following equation:
Find an expression for qm. For a practical device, two of these electrodes are used, connected in series. In operation, opposite charges of equal magnitude are stored on each electrode. Plot the differential and integral capacitance of the device as a function of potential. Comment on the degree of variation between the single electrode and the device.
11.11. Compare and contrast differences in cyclic voltammograms for capacitors and redox reactions.
11.12. Calculate the round-trip efficiency of an EDLC that is discharged and charged at a constant current of 13 A. The capacitance is 150 F with an effective ESR of 14 mΩ. The maximum voltage is 2.7 V and the discharge is terminated at 1.35 V.
11.13. Create the equivalent of Figure 11.16 for a nonaqueous electrolyte. Except for the information provided below, use the data from Illustration 11.6 and assume a voltage of 2.3 V. Compare the curves and physically explain the difference between aqueous and nonaqueous capacitors.
| Effective conductivity of electrolyte in separator | 1.1 S·m−1 |
| Capacitance per actual surface area | 0.100 F·m−2 |
| Effective conductivity of electrolyte in electrolyte | 0.7 S·m−1 |
| Density of electrolyte | 950 kg·m−3 |
11.14. A EDLC with a maximum potential of 4.2 V is charged at constant current (CC) until the potential reaches 4.2 V. The charge is then continued at 4.2 V (CV) until the current becomes very small. Finally, the cell is discharged at a current of 0.1 A, starting at t = 10 seconds. The data are shown in the figure, the inset shows the step change in current to 0.1 A for the discharge. Calculate the nominal capacitance of the EDLC and its ESReff from these data.
11.15. Below are Nyquist plots for two EDLC. The only difference is the loading of the electrode, which affects the thickness and capacitance of the device. What can be said about the conductivity of the solid compared to that of the electrolyte? What are the ESReff, ESR, and EDR for the device loaded to 11.3 mg cm−2? Which electrode loading would have a higher cutoff frequency? (Data adapted from Taberna et al. (2003) J. Electrochem. Soc., 150, A292.)
11.16. In Chapter 7, curves for the potential versus SOC for some common insertion electrodes used in Li-ion cells were presented. Lithium titanium disulfide is an insertion material where the potential varies linearly with the concentration of lithium between the titanium disulfide galleries. Considering the discussion in Section 11.8 on faradic reactions where potential changes with admitted charge, can you make the case that these insertion devices might equally well be called pseudo-capacitors?
11.17. In order to achieve a specific capacitance of 50 C·g−1, what minimum concentration of electrolyte is needed to avoid depletion? Assume a fully dissociated 1 : 1 electrolyte. The porosity is 0.65 and the density of the active material is 1800 kg·m−3.
11.18. One mechanism for self-discharge is caused by a faradaic reaction under kinetic control, perhaps from some impurity. If the faradaic reaction is controlled by Tafel kinetics, show that the leakage current depends on the logarithm of time. Assume the capacitance is constant.
11.19. Show that for an ideal capacitor in series with a resistance ESR, the maximum power is V2/4ESR. For an ideal battery, the maximum power is V2/4Rcell.
11.20. There are two common methods to measure leakage current. In the first, the potential of the device is measured at open-circuit over time. Explain how the V(t) data can be converted to I(t). Sketch how current would change over time. Why is it important to specify the time when leakage currents are reported?
11.21. Using the open-circuit data provided, what is the resistance associated with the leakage current for a 3 F capacitor? Estimate the 72-hour leakage current.
| Time [s] | Potential [V] |
| 0 | 2.800 |
| 600 | 2.799 |
| 1200 | 2.797 |
| 1800 | 2.796 |
| 2400 | 2.794 |
| 3000 | 2.793 |
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