In this section, we apply impedance spectroscopy to electrochemical double-layer capacitors in order to gain insight into their transient behavior. Additionally, we use the impedance results as a basis for a simplified EDLC model that will facilitate our analysis of these devices.
Analysis for Highly Conductive Solid Phase
In order to use impedance to examine the transient behavior of EDLCs, we begin again with Equation 11.24. This partial differential equation describes the potential as a function of time and position in a porous electrode assuming that the solid matrix is infinitely conductive. To obtain an expression for the complex impedance, we apply an oscillating potential at x = 0, as demonstrated in Chapter 6. The details are not shown here as they are a bit tedious, but the results are quite useful. Specifically, we obtain the following expression for the complex impedance (in ohms):
(11.27a)
where is a dimensionless frequency defined as the frequency, , divided by the characteristic frequency:
(11.27b)
The impedance described by Equation 11.27 is shown on a Nyquist plot, Figure 11.12, for three different electrode thicknesses. The parameters used to make this plot are included with the figure. First, we note that at high frequencies the impedance goes to zero. This is a consequence of assuming that the conductivity in the solid phase is infinite. At high frequencies, changes in potential do not penetrate significantly into the electrode and the effective resistance is zero. For a porous electrode, the slope of the impedance is 45° at high to moderate frequencies. As long as the time is short, the potential does not penetrate to the back of the electrode, and the electrode is effectively semi-infinite. As the frequency decreases, the potential penetrates farther and farther into the electrode. Eventually, the influence of the back of the electrode is felt. At frequencies sufficiently low that the capacitor has time to fully charge, the impedance becomes infinite as expected for an ideal capacitor. Note that the value for Zr where the line becomes vertical depends on the thickness of the electrode. That makes sense since both the capacitance and the time required to charge fully the porous capacitor increase with the electrode thickness.
Figure 11.12 Nyquist plot for EDLCs of varying thickness using Equation 11.27a.
The capacitance at low frequencies is directly proportional to the electrode thickness since all parts of the electrode are accessible. The low-frequency capacitance in farads is simply
(11.28)
Actually, it is the relationship of the frequency to the characteristic frequency that is important. Therefore, a more precise statement of the criterion is
(11.29)
The characteristic frequency is also called the cutoff frequency, . Below this frequency, the capacitance is nearly constant, and nearly all of the available electrode area is accessible. Under such conditions, Equation 11.28 describes the capacitance, and the full capacity of the electrode can be utilized. In contrast, at high frequencies, only a fraction of the available capacity is used. Generally, we’ll want to design or select a capacitor to operate below the cutoff frequency in order to use it as effectively as possible. Note that the cutoff frequency can equivalently be expressed as .
The capacitance of the electrode is given by
(11.30)
Using the same parameters from Figure 11.12, the capacitance is plotted in Figure 11.13 as a function of frequency. As noted above, the capacitance at low frequencies is constant and depends directly on the thickness of the electrode. At frequencies above the cutoff frequency, the capacitance drops quickly and goes to zero at high frequencies.
Figure 11.13 Capacitance of a 1 cm2 porous electrode. The thickness of the electrode is a parameter.
Referring back to Figure 11.12, we see that the impedance behavior roughly approximates that of an ideal capacitor (vertical line on Nyquist plot) in series with a resistance. This approximation provides the basis for a simplified series-RC circuit model as illustrated in Figure 11.14. The resistance is associated with the resistance to current flow in the electrolyte solution since we have assumed that the solid-phase resistance is zero (i.e., the solid is infinitely conductive). It is referred to as the equivalent distributed resistance (EDR). The value for the EDR can be expressed as
(11.31)
and is the limit of the real portion of the complex impedance as the frequency goes to zero. Note that this resistance in ohms is directly proportional to the thickness of the electrode, L. This simplified model is applicable when the frequency is below the cutoff frequency.
Figure 11.14 Comparison of porous electrode model (a) with simple model (b) at (1) ω = ω(circle), (2) ω = ω/3 (triangle), and (3) ω = 3ω* (square).
How does this simplified series-RC model compare with the behavior of the actual electrode? At high frequencies, the simple model overestimates the electrode resistance, since the resistance in the simple model is constant and that of the porous electrode is very small at high frequencies and increases with decreasing frequency as the penetration into the electrode increases. Therefore, the simple model will tend to underestimate the maximum power. However, at lower frequencies where nearly all of the electrode capacity is used, the simplified series-RC circuit model does a good job of representing the actual behavior. As noted previously, we typically design the capacitor to use the available capacity. Under those conditions, the simple model does a good job of representing the actual behavior of the system. Use of this approximation is shown in Illustration 11.5.
ILLUSTRATION 11.5
For the electrode described in Illustration 11.4, determine a simplified series-RC circuit model consisting of the EDR and an ideal capacitor. At what frequencies does this model apply?
The capacitance,
Thus, the simplified model is a 9 F capacitor in series with a 0.22 Ω resistor.
The cutoff frequency can be estimated as follows:
The frequency should be lower than this value for the full capacitance to be utilized. Now let’s compare the charging time from the simple model with the characteristic time that we calculated from the more complex model. For the simple model,
For a capacitor in series with a resistor (Equation 11.14 with Qi = 0) and a time of 6 seconds (see Illustration 11.4):
or 95% charged, which is consistent with the previous results, where charging required ≈6 seconds.
Analysis When Both Solid and Electrolyte Resistances Are Important
The results of the previous section are for the case where the conductivity of the solid phase is much larger than that of the electrolyte, a condition that is satisfied for many practical electrodes. If the resistances of both phases (solid and liquid) are important, the impedance can also be determined analytically:
(11.32a)
where
(11.32b)
Note that the dimensionless frequency, has the same form as it did before, but now utilizes a composite conductivity that includes the contributions of both the electrolyte and the solid. Figure 11.15 shows the impedance behavior for a 100 μm thick electrode at different ratios of κ/σ as a parameter. All of the other parameters are the same as those used in the previous section to create Figure 11.12. Hence, the results can be compared directly.
Figure 11.15 Nyquist plot for finite resistance in both the electrolyte and solid phases.
For an infinitely conductive solid phase (κ/σ = 0), the resistance for a porous electrode EDLC approaches zero at high frequencies, as we saw previously. These results are identical to those shown in Figure 11.12. As κ/σ increases, which is equivalent to a decrease in σ for these simulations since κ was kept constant, we see two marked differences in the impedance. First, Zr does not approach zero at high frequencies when both σ and κ are finite. In fact, it approaches the following limit:
(11.33)
The EDR is also impacted by a finite solid conductivity:
(11.34)
reflecting the additional resistance that results from the finite solid conductivity. As expected, Equation 11.34 is equivalent to Equation 11.31 as σ → ∞.
ILLUSTRATION 11.6
For the electrode described in Illustration 11.4, determine values for the simplified model consisting of the EDR and an ideal capacitor if the solid conductivity is finite and equal to 25 S·m−1. What is the cutoff frequency? How do these values compare with those from Illustration 11.5?
The capacitance,
Thus, the simplified model is a 9 F capacitor in series with a 0.356 Ω resistor.
The cutoff frequency can be estimated as
The frequency should be lower than this value for the full capacitance to be utilized. Now let’s compare these results with the results from Illustration 11.5. The capacitance is the same, since it is unaffected by the solid phase resistance. However, the finite solid conductivity leads to an increase in the EDR and, consequently, a decrease in the cutoff frequency since the higher resistance increases the time required for charge and discharge of the electrode. Therefore, this electrode will not cycle as quickly as the electrode from Illustration 11.5.
11.6 Full Cell Edlc Analysis
With the analysis of porous electrodes presented above and the resulting simplified models, we can now address a full EDLC cell. Incorporating the simplified representation for each of the two electrodes yields two capacitors and two resistors in series. In addition, there are other resistances not directly associated with the porous electrodes themselves that have not yet been included. Hence, we add an equivalent series resistance (ESR) to account for these additional resistances. The most important resistance included in the ESR is that of the separator between the electrodes. Resistances such as contact losses are also included in the ESR. We can now combine all of the resistances into a single effective resistance for the cell:
(11.35)
where the subscripts refer to the two electrodes. We can also use an equivalent capacitance to represent the capacitance of the two electrodes in series.We are now prepared to examine some interesting questions regarding EDLCs. For example, for a given electrode size (area), how thick must the electrodes be to achieve a particular capacity, and at what frequency can the device be operated and still maintain that capacity? Or, for a given capacity and required frequency of operation, what is the required area and thickness of the electrodes? Note that the specific area for capacitor materials is often expressed in area per mass. To use this number in the equations that we have developed, we also need to know the mass per volume of the material in an electrode. This and other concepts of interest are demonstrated in the following illustration.
ILLUSTRATION 11.7
An electrochemical double-layer capacitor is constructed from two porous carbon electrodes separated by a porous membrane. Please determine the capacity of the electrode, its effective resistance ESReff, and the maximum frequency at which it can be operated and still use its full capacity. Assume that the contact resistance is 0.4 mΩ.
Thickness of separator 25 μm
Effective conductivity of electrolyte in separator 55 S·m−1
Superficial (geometric) electrode area 30 cm2
Porosity of separator, ɛs 0.5
Electrode thickness 50 μm
Capacitance per actual surface area 0.13 F·m−2
Effective conductivity of electrolyte in electrode 35 S·m−1
Specific surface area of electrode 600 m2·g−1
Density of carbon 2050 kg·m−3
Porosity of electrode, ɛe 0.4
Effective electronic conductivity 104 S·m−1
The specific interfacial area, a, is where is the volume fraction of carbon in the electrode. Below the cutoff frequency, the capacitance of one electrode is
From Equation 11.7, the capacitance of the two electrodes in series is 7.2 F.
The conductivity of the solid phase is so high that we can neglect any contribution to the ESR from the solid carbon electrode and the resistance is simply
The maximum operating frequency for essentially full capacity will be lower than the cutoff frequency associated with the individual electrodes due to the influence of the additional resistance (separator and contact) that increases the time required to charge or discharge the electrode. To estimate the requested frequency, we use three times the RC time constant for the equivalent circuit, which is the time that it takes to reach 95% of the final value. The maximum frequency can be approximated as the reciprocal of this value.
The maximum voltage at which an EDLC can be operated is typically determined by the stability of the electrolyte and would therefore be specified. However, our simple model also allows us, for example, to estimate the change in voltage or charge as a function of time for a step change in the current, similar to what we did in Section 11.3.
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