Many battery researchers may find themselves initially resistant to AC impedance spectroscopy. This is because books like Bard's *Principles and Applications of Electrochemistry* and Cao Chunan and Zhang Jianqing's *Introduction to Electrochemical Impedance Spectroscopy* primarily rely on rigorous formula derivations. Today, we will try to avoid formulas and focus on analyzing AC impedance spectroscopy, especially its applications in lithium-ion batteries.
Electrochemical impedance spectroscopy is a relatively new electrochemical measurement technique. Although its development history is short, it has developed rapidly and is now increasingly being used in electrochemical fields such as batteries, fuel cells, and corrosion and protection.
Electrochemical impedance spectroscopy (EIS)
This involves applying a small-amplitude AC sinusoidal potential wave with a different frequency to an electrochemical system and measuring the change in the ratio of the AC potential to the current signal (the system's impedance) with the frequency ω of the sinusoidal wave, or the change in the phase angle f of the impedance with ω.
This diagram provides a more intuitive understanding. A small-amplitude sinusoidal potential signal is generated using a waveform generator and applied to the electrochemical system via a potentiostat. The output current/potential signal is converted and then used with a lock-in amplifier or spectrum analyzer to obtain the output impedance and its modulus or phase angle. By changing the frequency of the sinusoidal wave, a series of impedances, impedance moduli, and phase angles at different frequencies can be obtained. Plotting these values yields the electrochemical impedance spectroscopy (EIS) – this method is called electrochemical impedance spectroscopy. Because the perturbation signal is an alternating current (AC) signal, the electrochemical impedance spectroscopy is also called AC impedance spectroscopy.
Electrode induction (EIS) can be used to analyze electrode process kinetics, double layer and diffusion, and to study electrode materials, solid electrolytes, conductive polymers and corrosion protection mechanisms.
Basic idea – to treat the electrochemical system as an equivalent circuit
When studying an electrochemical system using electrochemical impedance spectroscopy (EIS), the basic idea is to treat the electrochemical system as an equivalent circuit. This equivalent circuit is composed of basic components such as resistors (R), capacitors (C), and inductors (L) combined in different ways, such as series or parallel connections. EIS allows for the quantitative determination of the magnitudes of these components, and by utilizing the electrochemical meaning of these components, the structure of the electrochemical system and the properties of the electrode processes can be analyzed.
We can treat an electrochemical system with an unknown internal structure as a black box. Inputting a perturbation function (excitation function) into the black box will result in an output response signal. The function used to describe the relationship between the perturbation and the response is called the transfer function. The transfer function is determined by the system's internal structure; therefore, by studying the transfer function, we can study the system's properties and obtain information about its internal structure. If the system's internal structure is a linear, stable structure, then the output signal is a linear function of the perturbation signal.
The meaning of G(ω) is determined by the different input signals.
Starting from this formula: Y/X=G(ω), in short, X is the input disturbance signal, Y is the output signal, and G is the result. Their frequencies are all ω. If X is current and Y is potential, G(ω) is defined as impedance, denoted by Z; if X is potential and Y is current, G(ω) is defined as admittance, denoted by Y. Obviously, impedance and admittance are reciprocals of each other, and they are collectively called impedance, denoted by G.
Admittance is a vector that varies with angular frequency ω (and impedance Z is also a vector). It is usually represented by a complex function of angular frequency ω (or general frequency f), i.e., Z = Z' + jZ”, where Z' is the real part and Z” is the imaginary part. The figure below shows a typical complex function graph.
Two electrochemical impedance spectroscopy
Electrochemical impedance spectroscopy involves measuring the ratio of a perturbation signal X to a response signal Y at different frequencies ω. This yields the real part, imaginary part, magnitude, and phase angle of the impedance at different frequencies. These quantities are then plotted into various curves to obtain an electrochemical impedance spectrum. There are two commonly used electrochemical impedance spectra: one is called the Nyquist plot, and the other is called the Bode plot.
The Nyquistplot uses the real part of the impedance as the horizontal axis and the negative imaginary part as the vertical axis. Each point in the plot represents a different frequency, with higher frequencies on the left (high frequency region) and lower frequencies on the right (low frequency region).
A Bodeplot consists of two curves: the horizontal axis represents the logarithm of the frequency, and the vertical axes represent the logarithm of the impedance magnitude and the phase angle of the impedance, respectively. Using either a Nyquistplot or a Bodeplot, the impedance of an electrochemical system can be analyzed to obtain useful electrochemical information.
Prerequisites for EIS measurement
An electrochemical system must meet the following three basic conditions to ensure that the measured impedance spectrum is meaningful.
Causality condition: The output response signal is caused solely by the input disturbance signal. In other words, there is a unique causal relationship between the measured signal and the disturbance signal; any other interfering signals must be eliminated. This condition is relatively easy to meet if sufficient attention is paid to controlling environmental factors (such as temperature) of the electrochemical system.
Linearity condition: A linear relationship exists between the output response signal and the input perturbation signal. Normally, the relationship between current and potential in an electrochemical system is not linear, but rather a nonlinear relationship determined by the system's kinetics. However, when a small-amplitude sinusoidal potential signal is used to perturb the system, the potential of the perturbation signal and the current of the response signal can be approximated as linear, thus approximately satisfying the linearity condition. Typically, the amplitude of the sinusoidal potential signal used as the perturbation signal is around 5mV, generally not exceeding 10mV.
Stability condition: A perturbation should not cause changes in the internal structure of the system, and the system should return to its original state after the perturbation stops. For reversible reactions, the stability condition is relatively easy to satisfy. For irreversible electrode processes, as long as the change on the electrode surface is not rapid, and the perturbation amplitude is small and the duration is short, the system can also recover to a state not far from its original state after the perturbation stops. This can be approximated as satisfying the stability condition. For very fast electrode reactions, or when the perturbation frequency is low and the duration is long, satisfying the stability condition is more difficult, thus making EIS research on fast irreversible reactions somewhat challenging.
Another finite condition is that the impedance or admittance values measured over the entire frequency range are finite.
Characteristics of EIS Measurement
Quasi-steady-state method: By using a small-amplitude sinusoidal potential signal to perturb the system, when measured near the equilibrium potential, anodic and cathodic processes alternate on the electrodes, with opposite effects. Therefore, even if the perturbation signal acts on the electrodes for a long time, it will not lead to the cumulative development of polarization or the cumulative change of the electrode surface state (the destructive effect on the electrode surface state is relatively small). Therefore, the EIS method is a "quasi-steady-state method".
Calculation simplification: Due to the linear relationship between potential and current, the electrodes are in a quasi-steady state during the measurement process, which greatly simplifies the mathematical processing of the measurement results.
Rich in information: EIS is a frequency domain measurement method with a wide frequency range that can be measured, thus obtaining more dynamic information and electrode interface structure information than conventional methods.
From simple to complex, disassembling the equivalent circuit
First, you need to understand the meaning of each basic component in the Nyquist diagram.
Resistance: A point on the horizontal axis (real part) on the Nyquist graph.
Capacitance: On a Nyquist graph, it is a straight line coinciding with the vertical axis (imaginary part).
An RC circuit consisting of a capacitor R and a capacitor C connected in series: On the Nyquist diagram, this is represented by a straight line intersecting the horizontal axis at R and parallel to the vertical axis.
A circuit in which the capacitor R and capacitor C are connected in parallel: On the Nyquist diagram, it is a semicircle with a radius of R/2.
Two typical EIS charge transfer processes controlled by EIS
If the electrode process is controlled by the charge transfer process (electrochemical reaction step), and the impedance caused by the diffusion process can be ignored, then the equivalent circuit of the electrochemical system can be simplified as follows:
Equivalent circuit: The charge transfer resistance is connected in parallel with the double-layer capacitance at the electrode-solution interface, and then connected in series with an ohmic resistor. The ohmic resistor includes the resistance of the solution in the measurement circuit. For a three-electrode system, it is the resistance of the solution between the working electrode and the reference electrode. For a two-electrode battery, it is the resistance of the solution between the two electrodes.
If we derive the formula, we can find that the resulting equation is the equation of a circle with center (RΩ+Rct/2, 0) and radius Rct/2, as shown in the figure below.
Rω and Rct can be directly calculated from the Nyquist plot, Zre = RΩ + Rct/2
Cd can be obtained from ω at the vertex of the semicircle, Cd = 1/ωR
It is important to note that:
In EIS measurements of solid electrodes, the curves always deviate from the semicircular trajectory to some extent, instead appearing as an arc, hence the term "capacitive arc." This phenomenon is known as the "diffusion effect." The cause of this diffusion is not fully understood, but it is generally believed to be related to the non-uniformity of the electrode surface, the adsorption layer on the electrode surface, and the poor conductivity of the solution. It reflects the property of the electrode double layer deviating from an ideal capacitance; in other words, simply equating the double layer at the electrode interface with a purely physical capacitance is inaccurate.
Solution resistance Rω includes not only the ohmic resistance of the solution, but also other possible ohmic resistances in the system, such as the ohmic resistance of the electrode surface film, the ohmic resistance of the battery separator, and the ohmic resistance of the electrode material itself.
EIS with mixed control of charge transfer and diffusion processes
If the charge transfer kinetics are not very fast, and the charge transfer and diffusion processes jointly control the overall electrode process, with electrochemical polarization and concentration polarization existing simultaneously, then the equivalent circuit of the electrochemical system can be simply represented as:
In addition to the charge transfer resistance, an impedance caused by the diffusion process is introduced into the circuit, denoted by Zω, and called the Warburg impedance. The Warburg impedance can be considered as a diffusion resistance Rω and a pseudo (diffusion) capacitance Cω connected in series.
After deriving the formula and drawing the graph, we can conclude that:
In the extremely low frequency region, diffusion control on the Nyquist plot appears as a straight line with a tilt angle of π/4 (45°).
In the high-frequency region, the equivalent impedance of the circuit when the charge transfer process is the control step is represented by a semicircle in the Nyquist diagram.
Therefore, on a planar electrode, when the electrode process is jointly controlled by charge transfer and diffusion, its Nyquist plot across the entire frequency domain consists of a semicircle in the high-frequency region and a 45-degree straight line in the low-frequency region (see figure below). The high-frequency region is controlled by electrode reaction kinetics (charge transfer process), while the low-frequency region is controlled by the diffusion of reactants or products of the electrode reaction. From the figure, the ohmic resistance, charge transfer resistance, double-layer capacitance at the electrode interface, and parameter s can be obtained. s is related to the diffusion coefficient, which can be used to estimate the diffusion coefficient D. Using the relationship Rct = RT/nFi0, the exchange current i0 of the electrode reaction can be further calculated.
Note: The above discussion is based on results obtained under the condition of semi-infinite linear diffusion with a planar electrode. In real systems, these conditions cannot be fully met, or other factors may influence the result; often, the diffusion impedance deviates from a straight line by 45 degrees, typically with a reduced tilt angle. This phenomenon arises from various reasons, but two main reasons are:
(1) The electrode surface is very rough, so the diffusion process is partly equivalent to spherical diffusion, as shown in this figure. The smaller the radius of the sphere, that is, the further it deviates from the flat plate electrode, the less the tilt angle of the straight line is than 45 degrees.
(2) In addition to the electrode potential, there is another state variable that causes inductive reactance during the measurement process.
For complex or special electrochemical systems, the shape of the EIS spectrum will be more complex and diverse, such as the possibility of two or more semicircles, or even semicircles in the second quadrant. In this case, resistors and capacitors alone are insufficient to describe the equivalent circuit, and other electrochemical components such as inductive reactance and constant-phase elements need to be introduced.
Meaning of impedance corresponding to each frequency band in the EIS of a lithium-ion battery
The extraction and insertion process of lithium ions in intercalation electrodes includes the following steps:
(1) Electrons are transported between active material particles, and lithium ions are transported in the electrolyte between the active material particles;
(2) Lithium ions diffuse and migrate through the insulating layer (SEI film) on the surface of the active material particles;
(3) Charge transport process at the electron/ion conductive junction;
(4) The solid-state diffusion process of lithium ions within the active material particles;
(5) The accumulation and consumption of lithium ions in the active material and the resulting changes in the crystal structure of the active material particles or the formation of new phases.
A typical EIS spectrum of lithium-ion deintercalation and intercalation processes in intercalation electrodes includes five parts:
(1) In the ultra-high frequency region (above 10kHz), the ohmic resistance related to the transport of lithium ions and electrons through the electrolyte, porous membrane, wires, active material particles, etc., is represented by a point on the EIS spectrum. This process can be represented by a resistance Rs.
(2) In the high-frequency region, a semicircle is associated with the diffusion and migration of lithium ions through the insulating layer on the surface of the active material particles. This process can be represented by a parallel RSEI/CSEI circuit. RSEI is the resistance of lithium ions diffusing and migrating through the SEI film.
(3) In the mid-frequency region, a semicircle is associated with the charge transfer process, which can be represented by a parallel Rct/Cdl circuit. Rct is the charge transfer resistance, or electrochemical reaction resistance, and Cdl is the double-layer capacitance;
(4) In the low-frequency region, there is a sloping line related to the solid diffusion process of lithium ions inside the active material particles, which can be represented by a Warburg impedance ZW describing the diffusion;
(5) The extremely low frequency region (<0.01Hz) consists of a semicircle related to the change in the crystal structure of the active material particles or the formation of a new phase, and a vertical line related to the accumulation and consumption of lithium ions in the active material. This process can be represented by a series circuit consisting of a parallel Rb/Cb circuit and a Cint circuit. Among them, Rb and Cb are the resistance and capacitance characterizing the change in the bulk structure of the active material particles, and Cint is the intercalation capacitance characterizing the accumulation or consumption of lithium ions in the active material.
The EIS test is generally performed in the frequency range of 10mHZ-10kHZ, with an amplitude of 5mV. Therefore, the resulting EIS graph is generally a focus on the real axis, i.e., the ohmic resistance Rs in (1), two semicircles or one semicircle, and a slanted line of about 45°.
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