As we noted in Chapter 5, a porous electrode is an effective means of increasing the surface area of an electrode. As with any porous electrode, the local resistance is a function of position in the electrode. An equivalent circuit diagram for an EDLC using a porous electrode is shown in Figure 11.10. The resistances along the top represent the electrolyte resistance through the thickness of the electrode. Along the bottom is another distributed resistance corresponding to that of the solid phase. The solid and electrolyte are connected by a capacitor associated with the double layer that is in parallel with a resistance, R_f, for faradaic reactions. For ionic current to travel to the back of the electrode, it must travel through the electrolyte. Current can also flow to charge the double layer along the thickness of the electrode. We now wish to examine the behavior of such a device and contrast it to a planar electrode. For our initial analysis we will assume that the effective conductivity of the solid phase is much larger than that of the electrolyte. , where  is a constant. We will then present more general results, but no attempt to derive them is made here. In the electrolyte we assume Ohm’s law applies, which means that we are also neglecting the influence of any concentration differences.

(11.20)

It is understood that the conductivity in Equation 11.20 is an effective conductivity that accounts for porosity and tortuosity of the electrode. In Chapter 5, the charge balance in the porous electrode assumed that only faradaic reactions occurred; both adsorption and charging of the double layer were neglected. The resulting balance was

(5.5)

For the EDLC, we must modify this charge balance. The actual current density at the electrode surface, i_n, can either be due to a faradaic reaction (through R_f in the figure) or to double-layer charging. When double-layer charging is accounted for, the charge balance is

(11.21)

where C_DL is the capacitance per unit area of electrode [F·m⁻²], and a is the specific interfacial area [m⁻¹], as introduced previously. There are clearly cases where both faradaic and nonfaradaic reactions are important, for example, in batteries, fuel cells, and pseudo-capacitors, but for now we neglect faradaic reactions. Furthermore, we consider only one spatial dimension, where the potential and current density are functions of time and position. With these assumptions, Ohm’s law and the charge balance reduce to

(11.22)

and

(11.23)

The current density can be eliminated from these equations to produce a single, second-order equation for the potential. To do this, Equation 11.22 is first differentiated with respect to z. We then multiply by –κ and add the result to Equation 11.23 to yield

(11.24)

Equation 11.24 may look familiar—it is similar in form to Fick’s second law of diffusion, which we encountered in Chapter 4. The equation is also analogous to the equation for a transmission line familiar from electrical engineering. Once the initial and boundary conditions are specified, there are numerous techniques available for solving this linear, partial differential equation.

Our objective is to gain quantitative insight into the time scale and penetration depth that characterize a porous EDLC electrode. In other words, we want to know how quickly we can charge an electrode of a given size or, conversely, the thickness of the electrode that we can effectively use in a given time. To do this, let’s consider a step change in potential at time equal to zero at the front of the electrode from ϕ_ini to a potential ϕ_f, for an electrode of thickness L. This process is equivalent to instantaneously charging (or discharging) the front of the electrode and then calculating the time it takes for the rest of the electrode to reach that same state of charge. Introducing the following dimensionless parameters:

where  has units of time. Equation 11.24 becomes

(11.25)

The initial condition is at τ = 0, θ = 0, with boundary conditions at x = 0, θ = 1 and at x = 1, . This last boundary condition is equivalent to setting the current density in the electrolyte to zero at the back of the porous electrode (i.e., at the current collector). The dimensionless potential from the solution of Equation 11.25 is plotted in Figure 11.11. From our result for the potential, we can calculate the current density at any position in the electrode from Ohm’s law. At very short times, the gradient in potential is very steep at the front of the electrode and the current density there is high. In contrast, the current does not penetrate very far into the electrode and is essentially zero for a large fraction of the electrode. Thus, for these short times, the electrode appears infinite, and the penetration depth can be approximated as

(11.26)

Equation 11.26 applies when semi-infinite conditions prevail, which is valid for , where the boundary condition at the back of the electrode does not impact the solution (see Figure 11.11).

Referring to the distributed capacitances in Figure 11.10, those near the front of the electrode charge or discharge more quickly because of lower resistance through the electrolyte due to a shorter transport distance. At short times, the current does not penetrate beyond a certain depth, and much of the electrode is left unused. Therefore, there is a limit to the thickness of the electrode that is effective for a particular application. Equation 11.26 can be used to determine the penetration depth corresponding to a given time associated with the application of interest, as shown in Illustration 11.4.

The group  has units of time and represents a characteristic time for charging the porous EDLC electrode. This characteristic time comes naturally from the above analysis, and corresponds to the time at which . As seen in Figure 11.11, it provides a reasonable estimate of the time it takes to charge the capacitor. Its reciprocal,  represents a characteristic frequency for the capacitor. The transient behavior of EDLCs will be explored in greater detail in the next section with use of impedance analysis, which builds upon the material that was introduced in Chapter 6.

ILLUSTRATION 11.4

Use the characteristic time to estimate the time required to charge a capacitor with the properties below, and then determine the penetration depth at 120 ms. Data for the electrode include the following:

κeff   = 15 S·m−1	σeff  ≫ κeff	L = 1 mm
a = 3 × 108 m−1	CDL = 0.3 F·m−2	A = 1 cm2

The time required to charge the capacitor is approximately 

The penetration depth at 120 ms can be estimated from Equation 11.26, since the time is less than τ/16.