Fundamentals
As mentioned previously, one of the key objectives of electroanalytical methods is to quantify the relationship between the current and voltage and to provide insight into the processes that influence that relationship. The simplest relationship is Ohm’s law:
which we have already used frequently. In this relationship, the resistance, RΩ, is the proportionality constant that relates current and voltage. Of course, the complex physical phenomena occurring in an electrochemical cell cannot be adequately represented with just a resistor. In particular, there are several physical processes that are time dependent: molecular diffusion, adsorption, and charging of the double layer, for example. The time constants associated with these processes vary significantly, making it possible to extract information about the system by examining the relationship between the voltage and current at different timescales. To do this, we need a method for describing and quantifying the relationship between current and voltage under dynamic conditions.
Here we briefly describe a technique that is known as electrochemical impedance spectroscopy or EIS. Our objective is to introduce you to the basic elements of the technique. For information beyond this initial treatment, please see references such as those listed at the end of the chapter. The electrochemical cell is the same as that described in Section 6.1, containing a WE, CE, and RE. Starting from steady state, the potential or current is perturbed by superposing an oscillating signal on the steady-state value. The response of the system to the oscillating input for a range of frequencies is then examined. Generally, this means that we look for the ratio of the voltage and current; this quantity is called the impedance, and has dimensions of ohm or ohm·m2.
Figure 6.16 illustrates the response of the current to an oscillating input potential. For small perturbations, the system is approximately linear and the current varies with the same frequency as the potential. The applied potential can be expressed equivalently as a sine or cosine wave, since both functions describe the same sustained oscillation. For our treatment, we use the cosine as follows:
where 
is the amplitude of the wave, f is its frequency [s−1], and 
[rad·s−1]. Note that in our discussion of impedance, the voltage, V(t), and the current, I(t), refer only to the oscillating portion and represent the deviation from the steady-state or baseline value. The current response that corresponds to Equation 6.18 has the same frequency, but may be shifted in phase (see Figure 6.16):
(6.19)
We seek the relationship between the voltage and the current, which depends on the physical characteristics of the system and changes with the frequency of the oscillating input. This relationship can be used as a “fingerprint” to characterize the physical system of interest.
EIS is performed by perturbing the experimental system with an oscillating input and measuring the corresponding output signal over a broad range of frequencies. The input and output signals are used to determine the impedance, which relates the current and voltage at different frequencies. The impedance data can then be fit to an equivalent circuit to provide quantitative physical insight into processes that control cell behavior at different timescales.
Impedance for Basic Circuit
In order to help you understand what the impedance is and how it represents some of the basic electrochemical processes that we have already identified, we will work this process “backwards,” beginning with the equivalent circuit shown in Figure 6.6, for which we have already gained some physical insight. This circuit represents the electrochemical interface as a capacitor (double layer) and resistor (charge-transfer resistance) in parallel. In series with the interface is a resistor that represents the ohmic resistance of the electrolyte.
In our case, we will use a voltage input and current output, although the reverse could also be considered with the same result. Because our simple circuit only has two types of circuit elements (resistor and a capacitor), we will begin with the more complicated of the two, the capacitor. The basic relationship between current and potential for a capacitor was introduced previously, and also applies for a system with a sustained oscillating perturbation (sine or cosine) of the voltage or current:
There are several ways to solve this equation for an oscillating input voltage. Here we will use complex variables in order to introduce the concept of the complex impedance that is most commonly used. As we saw above,
(6.18)
Alternatively, we could have chosen a sine wave
which would have yielded exactly the same relationship between the current and voltage (i.e., the same impedance). Because the system is linear, we can use superposition and add two inputs to get a combined output that is equal to the sum of the outputs from each individual input. Proportionality applied to the linear system allows us to multiply an input by a constant to yield a proportional output. Taking advantage of these properties, we define an alternative input that is equal to the sum of a cosine and sine wave,
where we multiplied the sine wave by the constant j, which is equal to 
. Our strategy is to solve the problem with 
as the input in order to simplify the math. Solution of the problem with 
as the input will provide a combined output. Once the combined output is determined, we can easily recover the desired portion of the output, which corresponds to only the cosine input, since the real (cosine) input will yield a real output, and the imaginary (sine) input will yield an imaginary output.
The reason why we have introduced this alternative input is to take advantage of Euler’s formula, which will simplify the math needed to solve this and more complex equations for circuit elements, and will lead naturally to the complex impedance:
(6.23)
With this formula, we now rewrite the input voltage from Equation 6.22 as
(6.24)
In this form, we can easily differentiate the complex input potential and use Equation 6.20 for a capacitor to determine the complex current response as follows:
As stated above, we can recover the time-dependent current that results from just the (real) cosine input by taking the real portion of our answer:
A key advantage of this approach is that it transforms derivatives with respect to time into algebraic terms, a fact that greatly facilitates solution. Equation 6.26 represents a solution to Equation 6.20 and provides the time-dependent current that results from the cosine input defined in Equation 6.18.
The above problem was readily solved for 
. However, we would like to be able to solve for the response of the complete circuit rather than for just one element of the circuit. An effective way to approach the circuit problem is to determine the impedance for each element of the circuit, and then to combine the individual impedances to get an equivalent impedance for the entire circuit. The complex impedance is defined as the ratio of the complex voltage and the complex current as follows:
where r is the magnitude of Z and 
is the phase angle (see inset). Note that the time dependence cancels out. Both the phase angle and the impedance are a function of the frequency, but not of time.
From Equations 6.25 and 6.26, the complex impedance of a capacitor is
Again, the time dependence cancels out since, for a linear system, it is the same in both the input and the output.
Now that we have the impedance for the capacitor, we need the appropriate values for the resistors to complete the circuit shown in Figure 6.6. For a resistor, the current and voltage are always in phase and there is no imaginary component to the impedance. Therefore, the impedance for a resistor is simply equal to the resistance:
(6.29)
One of the useful characteristics of impedances is that they add together like resistors. Making use of this, we can derive the equivalent impedance in ohms for the circuit (Figure 6.6) as follows:
The impedances for the faradaic reaction and the double-layer capacitor have been added as parallel resistors in a circuit (Problem 6.4), and the real and imaginary terms have been grouped. The equivalent impedance can be used to relate the voltage and current for the complete circuit, which is much more complicated than the response due to any individual circuit element. You should understand and be able to reproduce Equation 6.30.
COMPLEX VARIABLES
Impedance is typically described with complex quantities. A complex number is expressed as the sum of real and imaginary portions:
where 
. a is the real part of Z, Re{Z}; and b is the imaginary part of Z, Im{Z}. There are other ways of representing a complex number. Graphically, a complex number can be interpreted with a polar form:
r represents the magnitude of the number (modulus) and 
is the phase angle in radians (argument). The value Z is a point in the complex plane.
Use of Euler’s formula (described in text) permits us to express Z in exponential form as follows:
We used the equation for a capacitor to solve for its response to the oscillating input (Equation 6.26) and for its complex impedance (Equation 6.28). However, we need not go through this detailed process every time since the same circuit elements are frequently used. Table 6.4 summarizes the impedance for three common circuit elements.
Table 6.4 The Impedance of Common Circuit Elements
| Component | Behavior | Impedance |
| Resistor | V = IRΩ | RΩ |
| Capacitor | C = (Q/V) or ICdV/dt | 1/jωC |
| Inductor | V = Ldi/dt | jωL |
For a circuit comprised of these elements, we can simply use the impedances of the individual circuit elements from the table and combine them by adding them as we would resistors. The result is the equivalent impedance for the circuit such as that shown in Equation 6.30 for the circuit of Figure 6.6.
How can we graphically represent this complex impedance? One common way is a Nyquist plot, shown in Figure 6.17, where the imaginary portion, −Zi, is on the ordinate and the real part, Zr, is on the x-axis. Note that since the complex portion is frequently negative, −Zi rather than Zi is traditionally plotted on the imaginary axis. The shape of the plot that corresponds to the circuit in Figure 6.6 is a semicircle and is a common feature of EIS impedance spectra measured experimentally. The impedance for this circuit approaches 
as 
and 
as 
. Also, at the top of the semicircle, 
. Knowing this, we can estimate the value of these parameters from EIS data obtained experimentally. One shortcoming of the Nyquist plot is that frequency is not displayed explicitly on the diagram. The curve represents the whole range of frequencies, with high frequencies toward the left of the diagram and low frequencies toward the right.
It is useful to consider the physical significance of the behavior shown in Figure 6.17. At high frequencies, the capacitor acts as a “short” since the voltage oscillates at a frequency that is fast relative to the time required to charge the double layer. Because the double layer is never charged to any appreciable extent, the capacitor offers essentially no resistance to the oscillating current and the kinetic resistance, in parallel with the capacitor, does not play a role. Therefore, the complete circuit resistance is equal to the ohmic resistance. At lower frequencies where cycle time is comparable to the double-layer charging time, both the capacitor impedance and the faradaic (kinetic) resistance are important. The result is the semicircular region, with the phase shift caused by the capacitor. Finally, at sufficiently low frequencies (longer times), the cycle time is long relative to the double-layer charging time. Consequently, the double-layer charges and discharges quickly relative to the cycle time and does not contribute to the overall behavior; rather, essentially all of the current flows through the faradaic and ohmic resistors in series (see Figure 6.6). At these low frequencies, the circuit resistance for this simple circuit reaches a constant value of 
, and the output current is in phase with the input voltage. An understanding of EIS requires the ability to connect the physics, circuit model (equivalent circuit), and the complex impedance.
Connection of Complex Impedance to Time Domain
In the previous section, we derived an expression for the impedance of a capacitor and combined that expression with the faradaic and ohmic resistances shown in Figure 6.6 to determine the equivalent impedance for the circuit. We then demonstrated how to illustrate that complex impedance on a Nyquist plot. The semicircle that we obtained is a characteristic feature of impedance spectra for electrochemical systems and is connected to key physical parameters, as shown in Figure 6.17. In this section, we briefly examine the connection of the impedance to the time domain by using the complex impedance to determine the output current as a function of time.
The equations that we have used thus far are summarized below:
(6.31)
The process of converting from an arbitrary value of 
to the time-dependent expression for the current at that frequency is shown in Illustration 6.4. Its purpose is to show how the complex impedance relates both the magnitude and phase of the input and output signals. The data in the complex domain provide the information needed, as a function of frequency, to quantify the dynamic behavior of the system.
ILLUSTRATION 6.4
The complex impedance of an electrochemical system measured at a frequency of 1000 Hz was found to be 3 − j. The amplitude of the applied (oscillating voltage) is 5 mV. What is the corresponding current response in the time domain?
To answer this question, we first put Z = 3 − j into polar form 
.
Therefore, 
The magnitude of Z (equal to 3.16 in this example) is the ratio of the amplitudes of the voltage and current, 
. In other words, it tells you how much the current will change in response to a small change in the voltage, or vice versa. The angle 
(−0.322 rad in this example) provides the phase shift between the input and output. As illustrated by this example, the complex impedance, Z, provides both the ratio of the magnitudes of the voltage and current and the phase shift.
You may be thinking that 
is so small relative to the other term in the brackets that it will not make any difference. However, remember that cosine is a periodic function. The voltage and the resulting current are shown in the following figure, where the phase shift is evident.
Note that the voltage “lags” the current (reaches its peak at a time later than that at which the current reaches its peak).
Mass Transport
We now consider a common process for electrochemical systems, molecular diffusion, and how it might be included in our EIS analysis. Let’s start with one-dimensional diffusion where the process of interest is influenced by the concentration of a single species. The relevant equation is
Consider a semi-infinite domain with small perturbations around the initial uniform concentration, 
. The steady-state solution is just a constant; consequently, we are left to solve for the oscillating portion. Similar to what we did previously, we use a combination of cosine and sine terms to define the oscillating input:
where
Note that the steady-state value has been subtracted off in (6.34), as with other variables in Section 6.7. Equation 6.33 can be differentiated and substituted into (6.32) to yield
This ordinary differential equation can be solved for 
. However, we seek an expression for the complex impedance due to mass transfer. Considering Equation 6.27, we need to express both the time-varying voltage and current as a function of the oscillating concentration in order to get the desired impedance. For fast kinetics (quasi-equilibrium), the Nernst equation can be used to relate the change in voltage to the change in concentration at the surface. Furthermore, since the amplitude of the oscillation is small, the expression can be linearized to yield
The current is related to the flux at the surface as follows:
(6.37)
The impedance becomes
We can now solve Equation 6.35 for the required ratio of the surface concentration to the surface flux, which can then be substituted into Equation 6.38 to yield the desired impedance. The boundary conditions include Equation 6.36 for the concentration and 
. The result is
(6.39)
where Z is in ohms. This impedance yields a line with a 45° slope on a Nyquist plot at low frequencies (why?). It is most commonly called the Warburg impedance. To get the impedance in ohm·m2, which is what results if the current density rather than the current is used, the impedance 
should be multiplied by the area.
The Warburg impedance shown in Equation 6.40 assumed that the kinetic resistance was not significant, which permitted the use of the Nernst equation to relate the voltage and concentration. A complete kinetic expression (e.g., Butler–Volmer with concentration dependence) can alternatively be used with the transport equation to obtain the expressions for both the kinetic resistance and Warburg impedance. The interested reader should refer to a more advanced treatment such as the text by Orazem and Tribollet (2008) for additional information.
A procedure similar to that used to derive Equation 6.40 was also used to determine the complex impedance for mass transfer through a stagnant film of finite thickness δ to yield
(6.41)
where Z is again in ohms. The Nyquist plot for the finite thickness impedance with −Zi is on the ordinate and Zr is on the x-axis as shown in Figure 6.18, where the impedance has been made dimensionless by multiplying by 
. As mentioned previously, each point on the graph represents a different frequency, with high frequencies on the left. The impedance increases with decreasing frequency to yield the 45° slope characteristic of semi-infinite diffusion. The impedance levels off as the frequency decreases further owing to the finite diffusion layer thickness.
Another common representation of the complex impedance is the Bode plot, shown on the right of Figure 6.18. Here frequency is the independent variable and the magnitude and phase angle are plotted on the left and right axes. The key feature here is that the frequency or equivalently the characteristic time constant for different processes is highlighted by a change in magnitude and phase angle. For the diffusion example here, it is not surprising the change occurs at a dimensionless frequency of about 1, where the dimensionless frequency is 
.
Interpretation of Data
It is common to measure impedance data and then use software to fit the data to a variety of circuit elements. This approach is intuitive and generally helpful in correlating features of the impedance with physical phenomena in the electrochemical system. It also yields quantitative parameters characteristic of the system of interest. In this section, we explore further the relationship between the impedance data and the key parameters that characterize an electrochemical system. To do so, consider the circuit shown in Figure 6.6, but with a slight modification. Namely, we will add an element that accounts for diffusion, the Warburg impedance, as shown in Figure 6.19.
Finally, we note that the treatment of electrochemical impedance spectroscopy provided in this text is just an introduction. There are many important characteristics of electrochemical systems that influence their dynamic behavior that have not been considered, where additional complexity is required for accurate representation. For example, these characteristics may include the adsorption of species on the surface, the presence of surface films, or even a polymer coating on the surface. Frequently, the circuit representation of complex systems is not unique, and is best coupled with a physical description to enhance understanding. You should refer to more comprehensive resources for EIS analysis of such systems (see Further Reading section).
ILLUSTRATION 6.5
Suppose that you have an electrochemical system characterized by the following parameters
- CDL = 0.2 F·m−2
- Electrode dimensions: 5 cm × 5 cm (only one side active)
- Conductivity of electrolyte = 10 S·m−1
- Distance from electrode to reference electrode = 1 cm
- io = 10 A·m−2, single-electron reaction (n = 1)
= 1 × 10−9 m2·s−1- co = 10 mol·m−3
- T = 25 °C
Assuming that the system can be adequately represented by the circuit shown in Figure 6.19, create a Nyquist plot for the system.
SOLUTION:
In order to generate the Nyquist plot, we need to develop an expression for the complex impedance and then plot the real and imaginary portions of that impedance for a range of frequencies. The desired plot is in ohms, and involves the capacitance, solution resistance, Warburg impedance, and the faradaic resistance. The capacitance is
Based on Table 6.4, the complex impedance of the capacitor is
To determine the resistance of the electrolyte, we use Equation 4.8c:
For the kinetic resistance, we assume open circuit as the steady-state condition, with small oscillations around that point. Because the magnitude of the potential change is small, linear kinetics can be used to determine the resistance according to Equation 4.62:
The Warburg impedance is
We now have all that we need to assemble the composite complex impedance:
where the only unknown is the frequency. Perhaps the easiest way to generate the desired plot is to enter the above information into a package, such as Python or Matlab, that handles complex arithmetic, and calculate Z as a function of the frequency for a range of frequencies. We can then plot the real and imaginary parts to create the desired figure:
It is important to be able to solve this problem in reverse, that is, to go from measured data to parameters that represent the system. To practice this, use the above diagram and see which parameters you can extract. You may want to refer to Figure 6.17. Compare the values from the diagram with the values that we used to generate the diagram (see above).
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