A Brief Introduction to Impedance Spectroscopy

The objective of this book is to provide a comprehensive introduction to the measurement and analysis of impedance spectra. The objective of this chapter is to provide a brief and somewhat qualitative introduction to the subject.

Consider the system with unknown properties that is shown in Figure 2 and labeled black box. The objective of the exercise is to learn the properties of the box in an effort to understand what it is. A series of measurements may be considered to interrogate the black box by imposing an input and measuring the result. For example, imagine that the box is placed in a dark room and is then subject to light of a specified wavelength. If a response is seen, such as an electrical current, the box may be considered to be photoactive. To explore the kinetics associated with the discharge of current from absorbed photons, the light intensity could be modulated. Such an experimental approach is considered in Section 15.3.2 on page 438. An alternative approach could be to impose an electrical potential and observe the resulting current. Modulation of the input signal would allow exploration of the influence of storage of charge within the box and the kinetics of processes that transform the potential to current.

Figure 2: Representation of a black box.

The relationship between input and output is called a transfer function. Impedance spectroscopy is a special case of a transfer function.

Transfer Function

The transfer function provides a compact description of the input-output relation for a linear time-invariant (LTI) system. Due to the fact that most signals can be decomposed into summation of sinusoids via Fourier series, the response of a system is characterized by the frequency response of the system. A generalized system is illustrated in Figure 3(a). The response to a step input signal X(t) shows different long and short time behaviors that can be represented as the dependence of the transfer function on frequency.

Figure 3: Representation of the system response Y(t) to X(t) defined as: a) step change input; and b) sinusoidal input with frequency w.

The short-time behavior corresponds to high frequencies, and the long-time behavior corresponds to low frequencies. For an electrochemical system, charging of the electrode-electrolyte interface occurs rapidly and is associated with the high-frequency or short-time response. Diffusion is a slower process with a large time constant and correspondingly a smaller characteristic frequency.

Example 0.1 Characteristic Frequencies: Find the characteristic frequency for double-layer charging, faradaic reactions, and diffusion for a disk electrode with radius r0 = 0.25 cm and a capacity C0 = 20 µF/cm2. The disk is immersed in an electrolyte of resistivity ? = 10 Ocm. Assume that the faradaic reaction has an exchange current density i0 = 1 mA/cm2 and that the disk is rotating at O = 400 rpm. The kinematic viscosity of the electrolyte is v = 10T2 cm2/s.

Solution: Calculations are presented below for the time constants and characteristic frequencies associated with charging, faradaic reaction, and diffusion:

Double-Layer Charging: The time constant for charging the electrode surface is given by

where Re is the ohmic resistance, given for a disk electrode by equation (5.112) on page 115 in units of O as

or, in units of Ocm2,

For the parameters given, the time constant is tC = 0.04 ms. The corresponding characteristic angular frequency is given by

and, in units of Hz,

Thus, for a time constant tC = 0.04 ms, the characteristic frequency is 4.1 kHz.

Faradaic Reaction: The time constant for a faradaic reaction is given by

where Rt is the charge-transfer resistance, given for linear kinetics on a disk electrode by equation (5.117) on page 116, or

Equations (6) and (7) yield a time constant of 0.51 ms and a characteristic frequency of 310 Hz.

Diffusion: The time constant for diffusion to a rotating disk electrode is given by

where Di is the diffusion coefficient for the reacting species and dN is the diffusion layer thickness given as a function of rotation speed by equation (11.72) on page 262. The time constant for diffusion of a species with a diffusivity of 10T5 cm2/s is equal to 0.41 s. The corresponding characteristic frequency is 0.4 Hz.

Experiments are conducted in the time domain. If the input signal is sinusoidal, as shown in Figure 3(b),

where is the steady-state or time-invariant part of the signal, and |?X| represents the magnitude of the oscillating part of the signal. When |?X| is sufficiently small that the response is linear, the output will have the form of the input and be at the same frequency, i.e.,

where f is the phase lag between the input and output signals. An alternative representation of the time domain expressions is developed in Example 1.9 on page 20 as

and

respectively, where and are complex quantities that are functions of frequency but are independent of time. The transfer function is a function of frequency and is independent of both time and the magnitude of the input signal. While the measurements are made in time domain, the determination of the transfer function is obtained from subsequent analysis.

The calculation of the transfer function at a given frequency ? is presented schematically in Figure 4. The ratio of the amplitudes of the output and input signals yields the magnitude of the transfer function. The phase angle in units of radians can be obtained as

Figure 4: Schematic representation of the calculation of the transfer function for a sinusoidal input at frequency ?. The time lag between the two signals is ?t and the period of the signals is T.

If ?t = 0, the phase angle is equal to zero. Similarly, the phase angle is equal to zero if ?t = T. As shown in Figure 4, the output lags the input, and the phase angle has a positive value.

The transfer function is, therefore, characterized by two parameters: the gain

and the phase shift f(?). These two parameters can be written in the form of a complex number with a magnitude and a phase f(?) or with a real part expressed as cos (f(?)) and an imaginary part sin (f(?)) (see Chapter 1 on page 3).

Generally the input signal is considered to be a reference for the phase. In this case, the corresponding complex number for the input is real, i.e., and the output signal is a complex number with a magnitude and a phase f(?). Thus,

For an electrical or an electrochemical system, the input is usually a potential, the output is a current, and the transfer function is called admittance. In the particular case where the input is a current and the output is a potential, the transfer function is an impedance. The transfer function is, however, a property of the system that is independent of the input signal. As the admittance is the inverse of the impedance,

Generally only the impedance is considered even if the measurement corresponds to an admittance. The measured impedance can have a strong dependence on the applied frequency. By analyzing the impedance as a function of frequency, a transfer-function model could be defined which takes into account all time constants of the corresponding system.

Example 0.2 Impedance and Ohm's Law: How can impedance spectroscopy be differentiated from simple application of Ohm's law, i.e., V = IR?

Solution: The expression V = IR represents a steady-state measurement. For a system consisting of a resistor, shown in Figure 5(a), the measurement of current at an applied potential yields the value of the resistor, i.e., . For potentiostatic impedance measurements, application of an oscillatory potential

Figure 5: Electrical systems: a) a resistor and b) a resistor in series with the parallel combination of a capacitor and a resistor.

Remember! 0.2 Impedance is a complex transfer function that relates an electrical output to an electrical input.

yields a current

where f is the phase lag between the current and potential. As described in Example...