电化学阻抗法的原理

Introduction

This application report introduces the theory of EIS and tries not to elaborate on mathematical and electrical theory. If you still feel that the material presented here is difficult to understand, don't stop reading. Even if you don't know all the discussion, you can get useful information from this application report.

This application report consists of four main sections:

• AC circuit theory and representation of complex impedance values

• Physical electrochemical and circuit components

• Common equivalent circuit model

• Extract model parameters from impedance data

It is assumed that the reader does not have a circuit theory or an electrochemical basis. Each topic begins at a relatively junior level and extends to cover more advanced materials.

AC circuit theory and representation of complex impedance values

Definition of impedance: the concept of complex impedance

Almost everyone knows the concept of resistance. It refers to the amount of resistance to current in the circuit. Ohm's law (Equation 1) defines that the resistance is the ratio of voltage to current.

(1)

The application of Ohm's law is limited to only one circuit component - the ideal resistor. The ideal resistor has the following characteristics:

• Obey Ohm's law at any current and potential level.
• The magnitude of the resistance is independent of frequency
• The current and potential signals of the alternating current pass through the same phase of the resistor.

However, the characteristics exhibited by circuit components in reality are more complicated. These forces us to abandon the simple concept of resistance and instead use the more common circuit parameter, impedance. As with resistance, impedance is also a measure of the magnitude of current resistance, except that it is not limited by the features listed above.

Electrochemical impedance is obtained by measuring the current by applying an alternating potential to the circuit. Assuming that a sine wave potential excitation signal is applied, an AC current signal corresponding to this potential response. This current signal can be analyzed using the sum of the sinusoidal equations (Fourier series).
Electrochemical impedance is usually measured with a small excitation signal. Therefore, the response of the electrodes is non-linear. In a linear (or non-linear) system, the current corresponding to the sinusoidal potential signal is also a sinusoidal signal at the same frequency, except that the phase shifts (see Figure 1). More details will be described later.

（2）

E _t is the potential at time t, E ₀  is the amplitude, and ω is the angular frequency. The relationship between the angular frequency and the frequency is as shown in Equation 3.

（3）

In a linear system, the response signal I _t moves with the phase angle and the amplitude changes.

（4）

An expression similar to Ohm's law can calculate the impedance of the system, as shown in Equation 5. Therefore, the magnitude of the impedance is related to Zo and Φ.

（5）

The sine function E(t) is plotted on the X axis and I(t) is plotted on the Y axis. The result is shown in Figure 2. This ellipse is "Li Sha Yu Tu". Analysis of Lissajous diagrams on an oscilloscope is an accepted method of impedance measurement before analyzing impedance using advanced EIS instruments.

 

 

According to the Euler relationship (Equation 6),

（6）

The impedance can be expressed in a complex function. The potential is described by Equation 7, and the response current is Equation 8.

（7）

(8)

The impedance represents a complex function as shown in Equation 9.

（9）

Data Display

It can be seen from observation 9 that Z(ω) is composed of two parts: the real part and the imaginary part. With the real part as the X axis and the imaginary part as the Y axis, a Nyquist diagram as shown in Fig. 3 can be obtained. Note that the Y-axis is negative in the graph, and each point of the Nyquist graph corresponds to a frequency in the impedance. Figure 3 shows that the low frequency is at the right and the high frequency is at the left.

The impedance in the Nyquist plot can be described as the vector modulus |Z|. The angle between the vector and the X axis is the phase angle.

 One of the main drawbacks of the Nyquist diagram is that you cannot see the frequency corresponding to any point in the graph.

 

The Nyquist plot in Figure 3 is caused by the circuit in Figure 4. A semicircle is a characteristic of a time constant signal. Electrochemical impedance maps usually contain several semicircles. And often only see a part of the semicircle.

Another common graphical method is called the Bode diagram. The impedance is shown as the logarithm of the frequency as the X-axis, the absolute value of the impedance (|Z|=Z0), and the phase angle as the Y-axis.

The Bode diagram corresponding to the circuit in FIG. 4 is as shown in FIG. The Bode plot shows the frequency information.

Linearity of electrochemical systems

Circuit theory differs between linear and nonlinear circuits. Impedance analysis of linear circuits is much easier than nonlinear ones.

The definition of the linear system described below is from the article "Signals and Systems" by Oppenheim and Willsky:

An important feature of linear systems is additiveity. If the input is a weighted sum of multiple signals, the output is a simple superposition, that is, the weighted sum of the system's response to each signal. Expressed mathematically, the continuous function y1(t) of time is the response to x1(t), and y2(t) is the output of the response to input x2(t). If it is a linear system, then:

1)  y₁(t) + y₂(t)= x₁(t) + x₂(t)

2)  ay₁(t)= ax₁(t)

 

For a regulated electrochemical system, the input is voltage and the output is current. Electrochemical systems are not linear. Double the voltage does not necessarily correspond to twice the current.

However, Figure 6 shows how the electrochemical system approximates a linear system. Taking a small enough potential current curve is approximately linear.

In a typical EIS test, an alternating current signal of 1-10 mV is applied to the system. Under such a small potential, the system can be approximated as linear.

If the system is not linear, the current response will contain harmonics of the excitation frequency. The harmonic is a frequency that is an integer multiple of the fundamental frequency. For example, the frequency of the second harmonic is equal to twice the fundamental frequency.

Some researchers have used this phenomenon. Linear systems should not generate harmonics, so the presence or absence of significant harmonic response determines whether the system is linear. Other researchers have deliberately used larger excitation signals. They use harmonic response to estimate the curvature of the system's current-voltage curve

Steady state system

EIS measurements take a certain amount of time (usually up to several hours). The system under test must be a stable system throughout the EIS measurement time. A common cause of problems in EIS measurement and analysis is the instability of the system under test.

In fact, steady state systems are difficult to obtain. The test system varies with the adsorption of solution impurities, the growth of the oxide layer, the formation of reactants in the solution, the dissolution of the coating or the change in temperature, and some of the influencing factors are listed.

In an unsteady system, the EIS standard analysis tool may yield extremely inaccurate results.

Time, frequency domain and conversion

The signal processing theory refers to the data representation domain. The data displayed in different domains is the same. In the electrochemical impedance spectroscopy, two of the domains, the time domain and the frequency domain are used.

In the time domain, the signal diagram is shown as a signal amplitude vs. time plot. Figure 7 shows a signal diagram superimposed by two sine waves.

You can use one change to switch between the two domains. The Fourier transform converts the time domain into equivalent frequency domain data. A common term, fast Fourier transform, refers to a fast, computer-implemented Fourier transform. The inverse Fourier transform is the conversion of frequency domain data into time domain data.

<p? In modern electrochemical impedance systems, low frequency data is measured in the time domain. The computer applies a digitally approximated sine wave to the battery system via a digital-to-analog converter. The response current is measured by an analog-to-digital converter. The FFT is used to convert the current signal into a frequency domain.

The details of the transformation are not in this application report.

Circuit component

Electrochemical impedance data is usually obtained by fitting an equivalent circuit model. Most of the circuit components in the model are general purpose electrical components such as resistors, capacitors, and inductors. The components in the model should have physical electrochemical principles. For example, many models use resistors to simulate the solution resistance of a test system.

Therefore, knowledge about the impedance of standard circuit components is very useful. Table 1 lists the common circuit components, the relationship between voltage and current, and their impedance.

Component	Current Vs.Voltage	Impedance
resistor	E = IR	Z = R
inductor	E = L di / dt	Z = jωL
capacitor	I = C dE / dt	Z = 1/jωC

Note that the resistance value of the resistor is independent of frequency and has no imaginary part. With only the real part, the current remains constant across the phase and across the resistor.

The impedance value of the inductor increases as the frequency increases. The impedance of the inductor is only imaginary. Therefore, after passing through the inductor, the current is shifted by 90 degrees with respect to the voltage.

The impedance change of the capacitor is just the opposite of the inductance. The impedance value of the capacitor decreases as the frequency increases. The impedance of the capacitor is also only imaginary. Relative to the voltage, after the current passes through the capacitor, the phase is shifted by 90 degrees.

Series and parallel connection of circuit components

Few electrochemical systems can be simulated with a single equivalent circuit component. In contrast, electrochemical impedance spectroscopy usually has many components. The components are connected in series (Figure 9) and in parallel (Figure 10).

Some simple formulas can be used to represent the series and parallel connection of circuit component impedances.

(10)

For the series connection of linear element impedances, the equivalent impedance can be obtained by Equation 10.

(11)

For the parallel connection of the linear element impedance, the equivalent impedance value can be obtained by Equation 11.

(12)

Two examples are given to explain the problem of connection of circuit components. Assume that a 1 Ω and 4 Ω resistor is connected in series. The resistance value of the resistor is equal to the resistance value (see Table 1). Therefore, the total impedance value is calculated as shown in Equation 12.

When the resistors are connected in series, both the resistance value and the impedance value increase.

(13)

Assuming two 2μF capacitors in series, the total capacitance is 1μF.

When the capacitors are connected in series, the impedance value increases and the capacitance value decreases. This is due to the inverse relationship between the capacitance value and the impedance value.

Physical Electrochemistry and Equivalent Circuit Components

Solution resistance

Solution resistance is a significant cause of the impedance of an electrochemical test system. A potentiostat with a three electrode compensates for the solution resistance between the counter electrode and the reference electrode. However, the solution resistance between the reference electrode and the working electrode is also considered during testing.

 (14)

The electrical resistance of an ionic solution depends on the ion concentration, ion type, temperature and the area through which the current flows. In a bounded region, the area is A, the length is I, and the current flowing through is uniform, then the resistance is Equation 14.

(15)

ρ is the solution resistivity. The reciprocal of ρ (κ) is more common. κ is called the conductivity of the solution, and its relationship with the resistivity is as shown in Equation 15.

The standard chemistry manual usually lists the κ value of a particular solution. For other solutions, you can calculate the κ value by specific ionic conductivity. The unit of κ is S/m. Siemens is the reciprocal of Omega, so 1 S = 1/ohm.

Most electrochemical test systems are unevenly distributed when current passes through a certain solution area. Therefore, the main problems in calculating solution resistance include determining the current flow path and the geometry of the current flowing through the solution. A more comprehensive approach to calculating the actual resistance of a solution by ionic conductivity is not covered in this application note.

The solution resistance is usually not calculated by ionic conductivity. It can be obtained by fitting electrochemical impedance data.

Electric double layer capacitor

An electric double layer exists at the interface of the electrode and the solution. This electric double layer is formed by the adsorption of ions in the solution onto the surface of the electrode. The insulating space separates the charged electrode from the charged ions in the order of angstroms. The charge separated by the insulator forms a capacitance, and therefore, the surface of the metal immersed in the solution forms a capacitance. Although there are many factors affecting the electric double layer capacitance, the capacitance per square centimeter of the electrode surface is approximately in the range of 20 to 60 μF. Electrode potential, temperature, ion concentration, ion concentration, oxide layer, electrode surface roughness, impurity adsorption, etc. all affect the electric double layer capacitance.

Polarization resistance

When the electrode potential deviates from the open circuit potential, it is called polarization of the electrode. If an electric current flows through an electrochemical reaction occurring at the surface of the electrode, the electrode is polarized. Current flow is controlled by reaction kinetics and reactant diffusion.

When the electrode is uniformly corroded under an open circuit, the open circuit potential is controlled by the balance between the two electrochemical reactions. One of the reactions produces a cathode current and the other produces an anode current. When the cathode current and the anode current are equal, the open circuit potential reaches equilibrium. The open circuit potential is also referred to as a mixed potential. If the electrode is actively dissolved, the value of the current produced by either reaction is called the corrosion current.

Mixed potential control is also produced in systems where the electrode does not corrode. In the corrosion reaction discussed in this section, the terminology is modified so that it applies to the non-corrosive situation discussed later.

When only a simple, kinetically controlled reaction occurs, the potential of the electrode system is related to the following equation.

(16)

• I: Current measured by the electrode system • Icorr:
Corrosion current
• Eoc: Open circuit potential
• βa: Anode β coefficient (Tafel slope)
• βc: Cathode β coefficient

(17)

Applying a small potential signal, Equation 16 approximates Equation 17, which introduces a new parameter, polarization resistance Rp. In terms of name, the polarization resistance may behave like a resistor.

If the beta coefficient is known as the Tafel slope, Icorr can be calculated from Rp by Equation 17.

We discuss the parameter Rp in detail when discussing the electrode system model.

Charge transfer resistor

Similar resistance can be obtained from a single, kinetically controlled electrochemical reaction. In this case there is no mixed potential and only one reaction reaches equilibrium.

(18)

(19)

Assume that a metal matrix is ​​immersed in the solution. According to Equation 18 or more generally, the metal is ionized and dissolved into the solution.

When a forward reaction occurs, electrons enter the metal and the metal ions diffuse into the solution. The charge is shifting.

The charge transfer reaction has a certain speed. This speed is determined by the type of reaction, temperature, concentration of the reactants, and potential.

The relationship between potential and current is shown in Equation 20 (directly related to the number of electrons and Faraday's law)

 (20)

i0	Exchange current density
CO	Electrode surface oxidant concentration
CO*	Concentration of oxidant in bulk solution
CR	Electrode surface reducing agent concentration
or	Overpotential E app  – E oc
F	Faraday constant
T	temperature
R	Gas constant
a	Reaction constant
n	Molar number

(21)

When the ion concentration in the bulk solution is equal to the electrode surface, CO = CO* and CR = CR*. The formula 20 is simplified to the formula 21.

Equation 21 is called the Butler-Volmer formula. This formula only applies to polarization caused by charge transfer kinetics. Stir the solution to reduce the thickness of the diffusion layer and reduce the concentration polarization.

(22)

When the overpotential η is small, the electrochemical system is in equilibrium, and the charge transfer resistance expression is changed to Equation 22.

The exchange current density can be calculated when the charge transfer resistance is known.

 diffusion

Diffusion can also cause impedance, called Warburg impedance. This impedance depends on the frequency of the potential disturbance. At high frequencies, the Warburg impedance is small because the reactants do not have to diffuse too far. At low frequencies, the reactants need to diffuse far, causing the Warburg impedance to increase.

(23)

 

Equation 23 is the infinite Warburg impedance.

In the Nyquist plot, the Warburg impedance is a 45° oblique line to the X axis. In the Bode diagram, the Warburg impedance exhibits a phase shift of 45°.

In Equation 23, σ is the Warburg coefficient and is defined as Equation 24.

ω	Angular frequency
DO	Oxidant diffusion coefficient
DR	Reducing agent diffusion coefficient
A	Electrode surface area
n	Molar number

This form of Warburg impedance is effective if the thickness of the diffusion layer is infinite. However, this is not always the case. If the diffusion layer is finite (such as a thin cell or a coated sample), the impedance of the low frequency no longer follows the above equation. Instead, we get Equation 25.

 

d	Nernst diffusion layer
D	Average diffusion coefficient of diffused matter

This more common equation is called the limited Warburg impedance. The angular frequency tends to infinity at high frequencies, or δ tends to infinity in the infinite diffusion layer, tan h(δ(jω/D)1⁄2)→1, and Equation 23 is reduced to a finite Warburg impedance. Sometimes the equation is written in admittance form.

Coating capacitance

The two conductive plates are separated by a non-conductive medium to form a capacitor, a so-called dielectric. The size of the capacitor depends on the size of the panel, the distance between the panels, and the dielectric properties. The relationship is as shown in Equation 26:

 

e o	Vacuum dielectric constant
εr	Relative permittivity
A	Flat area
d	Distance between two plates

The vacuum dielectric constant is a physical constant and the relative dielectric constant depends on the material. Table 2 lists some useful e _r  values.

Table 2. Typical Dielectric Constants

Note the large difference in dielectric constant between water and organic coatings. When a coated substrate adsorbs water molecules, its capacitance changes, and the electrochemical impedance spectroscopy can measure its change.

Constant phase component

Capacitance in electrochemical impedance experiments is sometimes not ideal. Sometimes the surface is a constant phase element as defined below.

The impedance of the constant phase component is expressed as:

For constant phase components, the index α is less than one. The electric double layer capacitor in an actual electrochemical test system behaves like a constant phase element. Considering the non-ideal behavior of the electric double layer, several theories (surface roughness, leakage capacitance, uneven current distribution, etc.) have been proposed. Perhaps the best method is to treat α as an empirical constant with no physical meaning.

Virtual inductor

Sometimes the impedance of an electrochemical test system exhibits inductivity. Some researchers attribute the inductance behavior to the formation of a surface adsorption layer, like a passivation layer or a dirt layer. Others believe that the inductance is caused by test errors, including the unsatisfactory potentiostat.

Common equivalent circuit model

In the following sections we will introduce some common equivalent circuit models. These models can be used to explain simple electrochemical impedance data. Many of these models have been listed as standard models for the Gamry EIS300 electrochemical impedance spectroscopy software.

Table 3 lists the components used in the equivalent circuit. The equation for the admittance and impedance of each component is also given.

Table 3. Circuit Elements Used in the Models

The dependent variables used in these equations are R, C, L, Yo, B, and α. The EIS300 software package uses these dependent variables to fit the parameters.

Pure capacitor coating

The metal surface exhibits a high impedance when it is coated with an undamaged coating. Figure 11 shows the equivalent circuit in this case.

The model consists of a resistor (caused by a solution) and a series of coating capacitors.

Figure 12 shows the Nyquist plot of the model.

The capacitance value cannot be obtained from the Nyquist plot. It can be obtained by curve fitting or confirmation of data points. Note that the solution resistance can be estimated from the intercept of the curve from the real axis.

Figure 13 shows the corresponding. The capacitance can be estimated from the figure but the solution resistance value is not obtained. Even when the frequency reaches 100 kHz, the coating resistance value is much larger than the solution resistance.

Battery impedance webinar

On February 12th, 2014, Dr. Burak Ulgut successfully hosted a webinar on electrochemical impedance spectroscopy and its application in battery analysis. The relevant content of this seminar can be purchased at PlugVolt .