Recall from Chapter 3 that the interface between an electrode and the electrolyte is generally charged. There can be excess positive or negative charge in the metal that is balanced with an equal and opposite charge in the electrolyte adjacent to the surface. The counterbalancing charge may consist of adsorbed ions in the inner Helmholtz plane (IHP), solvated ions in the outer Helmholtz plane (OHP), and the ions in the diffused part of the double layer. This charge separation across the interface is effectively a capacitor and is capable of storing energy. As with other types of capacitors, work is required to achieve this charge separation, and work or energy can be extracted by eliminating this charge separation through the flow of current. Of course, there is a second electrode, not shown in Figure 11.2, that is required to create an EDLC as we will discuss shortly. Note that the separation of charge across the interface in an EDLC shown takes place without faradaic reaction (oxidation or reduction). In other words, there are no electrons transferred across the double layer and no change of oxidation state for the participating ions. In contrast, almost all of the current flow that we have discussed previously has been due to faradaic reactions. This distinction will be emphasized throughout the chapter, and an electrode that can be polarized (by applying a potential) without passing a faradaic current is called an ideally polarizable electrode.

The double layer does indeed provide charge separation and is capable of storing energy. But how can we achieve the charge separation depicted in Figure 11.2? We need a way to get charge on and off the electrodes. This is accomplished by adding a second electrode and by passing current between the two electrodes, which is equivalent to moving charge (electrons) from one electrode to the other. As electrons move through the external circuit, ions move in solution to balance the resulting charge on each electrode. This process is illustrated in Figure 11.3. Note that, as expected, a power supply is required for charging an EDLC. With a positive current, I, flowing through the leads as shown, electrons move in the opposite direction. An excess negative charge builds up in the left electrode, and an excess positive charge forms on the right electrode. To balance these charges, cations from the electrolyte move toward the electrode on the left; simultaneously, anions move to the right side to compensate for the missing electrons from that electrode. The movement of ions constitutes a flow of current in the electrolyte equal to that in the leads. The net effect is to have two double-layer capacitors in series, and the capacitance of the device is calculated with Equation 11.7 for capacitors in series.

Classical Descriptions of the Electrical Double-Layer Capacitance

As noted previously, the charge in solution can be found at multiple locations; therefore, the capacitance of an EDLC is comprised of multiple parts. Restricting ourselves to a single electrode, its capacitance is divided into three parts: the diffuse layer (GC), ions in the compact layer near the electrode (OHP), and ions adsorbed on the surface (IHP). The contribution of each of these will be discussed separately, but it is important to recognize that the individual contributions are not really independent.

The capacitance per unit area associated with the diffuse layer is denoted as C_GC, where the subscript recognizes Gouy and Chapman for their independent contributions to the theory. The capacitance associated with the charges in the diffuse part of the double layer for a symmetric electrolyte is

(11.8)

where λ is the Debye length. A symmetric electrolyte is one where a neutral salt dissolves into two ions of opposite charge, . For example, NaCl and ZnSO₄ are symmetrical electrolytes. Equation 11.8 was developed by Gouy and Chapman using reasoning similar to that used by Debye and Hückel for activity coefficients (see Chapter 2), although Gouy and Chapman’s work predates DH by about a decade. The mathematical approximation used by DH for the spherical geometry is not needed here. Figure 11.4 shows the capacitance of a 1 : 1 symmetric electrolyte as a function of potential. The concentration of ions in solution is shown as a parameter in the figure. There are two points of note. First, C_GC is a strong function of potential. As the potential moves away from the point of zero charge (PZC), the capacitance increases exponentially. Second, we see that the capacitance increases with concentration. Recall that the Debye length is given by

(2.39)

and provides a measure of the thickness of the diffuse layer. Physically, the amount of charge in the diffuse layer found close to the electrode increases as λ decreases, resulting in increased capacitance. Experimentally, this effect is shown in Figure 11.5 for a NaF solution using a Hg electrode. Observe that the hyperbolic cosine behavior is present at lower ionic strengths. Sodium fluoride was used because F⁻ ions are only weakly adsorbed. A Hg electrode might seem an odd choice; indeed, if our objective were to make a practical device, we would not choose mercury. However, much of the fundamental work with capacitance uses liquid Hg because of its ability to provide a reproducible, polarizable surface. Hg is also a good electrical conductor at room temperature.

As will be discussed in more detail later, high electrolyte concentrations are desirable for EDLC devices in order to minimize resistance and ensure that there are sufficient ions in the electrolyte reservoir to balance the charges on the electrode (see Figure 11.3). At high concentrations, the contribution to the capacitance from the diffuse layer suggests a very large value for the capacitance, and one that grows exponentially with potential. Experimentally this is not observed—a typical value for the capacitance associated with the double layer per unit of real surface area is 0.1–0.5 F·m⁻², and it is relatively constant. Thus, the GC model alone does not adequately represent the double-layer capacitance, even though it does show the correct variation of capacitance with concentration near the PZC at low concentrations. This deficiency of the GC model was addressed by Stern, who recognized that charges have a finite radius and cannot get arbitrarily close to the electrode surface, in contrast to the assumptions of the Gouy Chapman model. Accounting for this, the capacitance of the double layer in the absence of specific adsorption is traditionally expressed as

(11.9)

where d is the distance of closest approach for the ions. We can think of the two terms as defining (i) the capacitance of a compact layer of charge at the outer Helmholtz plane, C_OHP, originally described by Helmholtz, and (ii) that of the diffuse layer, C_GC. We further note that these two capacitances are typically combined as capacitors in series, as reflected in Equation 11.9 (refer also to Equation 11.7). The overall capacitance, C_DL, is dominated by the smallest capacitance in 11.9. At low concentrations, the smallest capacitance is C_GC, which controls the overall behavior. However, at the higher concentrations used in EDLC devices,

(11.10)

Therefore, for most practical systems, the capacitance associated with the diffuse layer is not important, and behavior is controlled by C_OHP.

We will now examine the capacitance associated with ions located at the OHP in a bit more detail. Once more, a physical picture of the structure of the double layer is key. Figure 11.6 presents a view of the double layer that focuses on the ions at the OHP. Here the surface of the electrode is envisioned to be completely covered with the solvent; for the moment, we are still not considering unsolvated ions that are directly adsorbed on the surface. The OHP is defined by solvated ions that are as close to the surface as possible while maintaining their waters of hydration. Clearly, even our simplified picture is far more complex than the normal parallel plate capacitor; nonetheless, we will see that Equation 11.9 gives a reasonable approximation for the capacitance. Two key parameters are d, the distance for charge separation, and the dielectric constant, ɛ_r. The dielectric constant of bulk water is about 80. The high relative permittivity of water is attributed to its polar nature and the ability of bulk water molecules to reorient freely under the influence of an electric field. This bulk value is no longer appropriate near the surface where movement is constrained. This is particularly true for the water molecules adjacent to the electrode surface (see Figure 11.6). We account for this by dividing the region between the OHP and the electrode surface into two parts with different dielectric constants, both of which are lower than the bulk value. C_OHP is approximated as (see Problem 11.5).

(11.11)

where r_w is the radius of the water, and r_i is the radius of the unsolvated ion. ɛ_rL is the low dielectric constant applicable for the water against the electrode surface, and ɛ_rH is a higher value appropriate for the intermediate region between the inner region and the OHP. Once again, these contributions are in series, and generally the first term dominates. Using a value of 6 for the low dielectric constant and 0.14 nm for the radius of water,

(11.12)

The magnitude of this number is in the range of typical experimental values measured for the double-layer capacitance.

As noted previously, except for very dilute solutions near the PZC, C_OHP is much smaller than the value of C_GC calculated for the diffuse layer with Equation 11.8. Based on this, we would expect the double-layer capacitance to be more or less constant, and largely independent of the nature of the electrode surface and the type of ions in solution. The highest capacitance would result if the excess charge in solution, that which balances the charge on the electrode, were concentrated at the OHP. As the distance between the charge and the surface increases, we expect that the capacitance would decrease.

We will now consider the influence of ions that have shed their waters of hydration and are specifically adsorbed on the electrode surface. These ions define the IHP as described in Chapter 3. Development of the theory will not be attempted here; instead, key physical features that are important for understanding the influence of adsorbed ions on the capacitance are identified. To illustrate these features, we will use experimental results, which are often presented in the form of the differential capacitance defined previously (Equation 11.2).

In the end, differential capacitance curves can be quite complex. A representative, but idealized, curve for the differential capacitance of an electrode is shown in Figure 11.7. The magnitude of the capacitance is approximately what we would expect from Equation 11.12. However, as we move away from the PZC, the capacitance increases due to ion adsorption in the IHP. The increase is generally not symmetric about the PZC; rather, it increases more rapidly at positive potentials. The asymmetry is caused by the fact that negative ions are more likely to adsorb than positive ions.

What is the effect of ion adsorption? It is tempting (but wrong) to simply add the capacitance associated with the adsorbed ions, C_IHP, as another capacitance in series with the C_GC and C_OHP, similar to what was done in Equation 11.9. A simple electrical series addition of capacitance attributed to adsorption could only result in a decrease in capacitance, never an increase. Clearly, the physical picture is more complex, and extending the above analogy is not fruitful.

Some insight into the problem can be gained from a charge balance. To maintain overall electroneutrality at the interface, the charge per unit area on the metal, q_m, is counterbalanced by the sum of the charges (i) in the adsorbed layer, q_IHP, (ii) at the outer Helmholtz plane, q_OHP, and (iii) residing in the diffuse layer, q_GC.

(11.13)

As discussed in Chapter 3, ions (particularly anions) can adsorb directly onto the metal surface, forming the IHP. For a given charge on the metal, q_m, the presence of charge at the IHP means that there is less charge at the OHP and beyond. Consequently, charge storage at each location in the double layer is not independent. As the potential is increased in either the positive or negative direction from the PZC, the fraction of ions located at the IHP tends to increase. An increased fraction of ions at the IHP leads to increased capacitance since the ions that adsorb are very close to the electrode surface (i.e., the distance d is small).

The adsorption of ions at the IHP is influenced by several factors that include both chemical and coulombic interactions between ions, the solvent, and the surface. In general, anions are larger, form weaker interactions with the solvent, and are more likely to adsorb than cations. As expected, a positive q_m facilitates anion adsorption, while a negative value promotes the adsorption of cations. Because anions are more readily adsorbed, the differential capacitance curve tends to be asymmetric, with higher capacitance at positive potentials for a given potential difference from the PZC, owing to increased ion adsorption. However, the surface charge is not the only important factor as demonstrated by the fact that, contrary to our intuition, it is possible for negative ions to adsorb onto a negatively charged surface. Also, lateral interactions between ions become increasingly important as the surface of the electrode is more and more populated with adsorbed ions, making it increasingly difficult for new ions to adsorb. It is the interaction of multiple competing factors that leads to the complex structure of the differential capacitance curves for ELDC electrodes, with features such as the hump shown in Figure 11.7 for a relatively simple curve.

In this section, the physical phenomena responsible for the observed capacitance in EDLCs have been discussed in the context of classical double-layer theory. The resulting potential dependence of the differential double-layer capacitance is markedly different from that of a conventional plate capacitor. The magnitude of the double-layer capacitance has been estimated and justified in terms of the relevant physical processes. This information provides a background for understanding the performance of ELDCs.

Capacitance for Practical EDLC Analysis

Since we expect the potential of a practical device to be operated over the full range of stability allowed by the electrolyte, how important is it to include the complete potential-dependent differential capacitance for the EDLC? Such a treatment would greatly complicate our analysis. Therefore, for practical engineering calculations, we will assume that the C_DL is constant. While not perfect, this simplification allows us to reasonably approximate the behavior of the full EDLC. An EDLC has two electrodes, and charge is stored in the double layer of each of these electrodes. The total capacitance of the device is

(11.14)

since the two double-layer capacitors are in series. One of the electrodes will have a positive charge, and the other a negative charge of equal magnitude. When the charge on the positive electrode increases, the charge on the negative electrode must also increase by an equal amount. Frequently, these two electrodes are identical, and we can use the same differential capacity curve for each of them. The asymmetry in the differential capacity curve implies that the electrode with the lower differential capacity will dominate. Thus, the fact that the two capacitances are in series reduces the variation of the overall capacitance with potential in the complete device. Also, the behavior of a device where the two electrodes are identical will be independent of the polarity to which it is charged. Finally, the practicing engineer is generally interested in the integral capacitance, where the variations in the differential curve are smoothed out. These behaviors are explored more fully in Problem 11.10.

As a final point related to practical devices, we note that a key advantage of the EDLC comes from using an electrode material with a high surface area. Carbon, for example, can have a surface area greater than 1000 m²·g⁻¹. A porous electrode, as described in Chapter 5, is a good way to achieve a large interfacial area between the electrode and electrolyte. Thus, in Figure 11.1c, we represent the two electrodes as porous electrodes. A typical value for the capacitance associated with the double layer per unit of real surface area is 0.1–0.5 F·m⁻². The capacitance of an electrode can be found by multiplying the electrode area by the capacitance per unit area. The total capacitance for the device would be approximately half of the value calculated for the single electrode as per Equation 11.7.

NOMENCLATURE

We will restrict ourselves to using the terms electrochemical double-layer capacitor (EDLC) and pseudo-capacitor (discussed in Section 11.8), where the names generally reflect the underlying physics of the devices. Within the scientific literature as well as product brochures, other commonly encountered terms are “supercapacitors” and “ultracapacitors,” but there appears to be little consensus as to the precise features associated with these terms.

ILLUSTRATION 11.2

For a 1 F capacitor, calculate the area required for each type of capacitor:

1. Conventional capacitor where the plates are separated by an insulator (ɛ_r = 100) with a distance of 50 μm between plates.
2. Electrolytic capacitor with foil gain of 25, dielectric constant of 10, and thickness of 225 nm.The area calculated here is the superficial area of the electrode; the true surface area is 25 times larger. Note that geometric area is also commonly used in place of superficial area.
3. EDLC, 25 μm thick carbon electrode with a porosity of 0.5, ρ_c = 2000 kg·m⁻³. The carbon has a specific surface area of 1000 m²·g⁻¹, with a capacitance of 0.15 F·m⁻² of carbon surface area.Therefore, the superficial area needed for a 1 F, ECDL capacitor is 2.7 cm². This low area is mostly because the actual area per unit volume is so high.