Electrochemical Impedance Spectroscopy (EIS) is a non-destructive technique used to analyze the electrical properties of an electrochemical system across a range of frequencies. It measures the impedance response of the system to an applied alternating current (AC) signal.

**Principle:**

EIS involves applying a small amplitude AC signal to the electrochemical cell and observing the resulting current response. By varying the frequency of the AC signal, EIS obtains impedance data across a spectrum. This technique analyzes the system’s resistance, capacitance, and other electrochemical properties, providing insights into its behavior at different frequencies.

**Electrochemical Impedance Spectroscopy (EIS)** operates on a three-electrode system to examine the electrical properties of an electrochemical interface across varying frequencies. EIS applies a small alternating current (AC) signal to the working electrode while the reference electrode maintains a steady potential. By monitoring the resulting current response at different frequencies, EIS offers insights into the system’s impedance behavior. This technique involves systematic variation of the applied AC signal’s amplitude and frequency, measuring the in-phase and out-of-phase current response. The collected data is plotted on Nyquist and Bode plots, illustrating impedance against frequency and impedance magnitude with phase angle, respectively. These plots aid in understanding charge transfer processes and the system’s electrical characteristics. Furthermore, fitting the impedance data to equivalent electrical circuits enables the extraction of electrochemical parameters, such as resistance and capacitance, shedding light on the electrochemical properties of the interface. EIS serves as a valuable tool for characterizing interface behavior, analyzing reaction kinetics, and comprehensively assessing the electrical aspects of electrochemical systems.

**EIS Parameters: **

**Quiet Time:**

- Duration of application of a DC bias potential before commencing the frequency scan.

**Starting Frequency:**

- The initial frequency of the potentiostatic sine wave applied in the experiment.

**Ending Frequency:**

- Final frequency of the potentiostatic sine wave applied during the experiment.

**Steps per Decade:**

- Quantity of potentiostatic sine waves applied per logarithmic frequency change per decade.

**DC Bias Potential: **

- Baseline potential was applied consistently throughout the experiment.

**Amplitude of AC Signal:**

- The magnitude of the applied alternating current signal used to perturb the electrochemical system.

**EIS Analysis:**

**Nyquist Plot and Bode Plot:**

EIS data is commonly represented in Nyquist and Bode plots. Nyquist plots display impedance against frequency and offer information about charge transfer processes, while Bode plots show impedance magnitude and phase angle as functions of frequency.

When a resistor and a capacitor are connected in parallel, an intriguing phenomenon emerges. Current follows the path of least impedance, irrespective of whether it’s an alternating current (AC) or direct current (DC). The capacitor’s impedance varies with frequency, altering the chosen path for the current.

At high frequencies, the capacitor’s impedance drops significantly, allowing most of the current to pass through it. As the frequency decreases, the capacitor’s impedance rises, causing a larger portion of the current to flow through the resistor. When most of the current passes through the resistor, the total imaginary resistance (Z’’) decreases while the real part (Z’) increases.

These changes result in a semicircular pattern on the Nyquist plot. It’s important to note that the Nyquist plot represents the complex plane where each value is a complex number, thus requiring proportional axes. When an ideal capacitor is in parallel with a resistor, this configuration precisely generates a semicircle on the Nyquist plot.

**Standardized Circuit Models**

Electrochemical Impedance Spectroscopy (EIS) data is typically subjected to analysis through fitting into an equivalent electrical circuit model. This model predominantly integrates conventional electrical components like resistors, capacitors, and inductors. The efficacy of these elements within the model relies on their alignment with the physical electrochemistry of the system. For instance, a prevalent inclusion in many models is a resistor that symbolizes the solution resistance within the cell.

A fundamental understanding of the impedance properties associated with standard circuit components becomes notably beneficial. The following details outline these common circuit elements, encompassing their current-voltage relationship equations and their respective impedance characteristics.

**Resistor:**

In the graph, the red peaks represent voltage peaks, while the (dotted lines) green peaks represent current peaks, When the potential begins at zero, the current also initiates from zero. At maximum potential, the current reaches its maximum, illustrating a synchronous relationship between them without any phase difference. Changing the frequency from 1 Hz to 4 Hz results in four cycles, while the peak values of the current remain constant. Consequently, the current remains in phase with the potential.

For a resistor, the current consistently aligns with the potential, maintaining a phase difference of zero. Notably, the current’s magnitude remains unaffected by changes in frequency.

**Capacitor:**

In the graph, the red peaks represent voltage peaks, while the (dotted lines) blue peaks represent current peaks, distinctly illustrating a phase disparity. In this representation, the current leads the voltage by 90°.

The relationship between current and voltage for a capacitor is delineated by a differential equation. In the case of AC, it is expressed as the product of capacitance (C) and the derivative of voltage concerning time:

The current leads the voltage by 90°.

**Inductor:**

The inductor exhibits a phase shift where the current lags the voltage. Similarly, to a capacitor, the difference lies in the direction of this lag; here, the current lags rather than leads as observed in capacitors. At 1Hz frequency, it demonstrates a specific magnitude. However, when a 3Hz frequency is applied, the magnitude diminishes. For an inductor, as the frequency increases, the number of currents decreases.

Therefore, the phase difference for an inductor is a 90° lag. The current lags the voltage by 90°.

**Constant Phase Element(Q):**

In Electrochemical Impedance Spectroscopy (EIS), capacitors don’t always adhere to ideal behavior. Instead, they can exhibit characteristics akin to a Constant Phase Element (CPE), described by an impedance equation of the form:

Where:

ZCPE is the impedance of the Constant Phase Element.

ω represents the angular frequency.

Y0 corresponds to a constant (like capacitance for an ideal capacitor).

α denotes the exponent, which differentiates a CPE from an ideal capacitor. For a standard capacitor, (α= 1), but for a CPE, (α < 1).

The “double-layer capacitor” observed in real cells often demonstrates behavior akin to a CPE rather than a conventional capacitor. While various theories attempt to explain this non-ideal behavior, a universally accepted explanation is yet to emerge. Consequently, α is often treated as an empirical constant, allowing for the characterization of the impedance without delving deeply into its physical underpinnings.

This empirical approach facilitates the practical use of EIS without requiring a detailed understanding of the physical basis of α in these non-ideal capacitive systems.

**Warburg (W):**

The Warburg impedance in electrochemical systems arises due to diffusion limitations encountered by reactants. It can be described by two equations: the “infinite” Warburg impedance and the more general “finite” Warburg impedance.

The “infinite” Warburg impedance, represented by Equation (5), is applicable when the diffusion layer is theoretically infinite in thickness. This equation involves a Warburg coefficient (σ) determined by factors such as the diffusion coefficients of the oxidant and reductant, the electrode’s surface area, the number of electrons involved, and the radial frequency (ω) of the potential perturbation.

However, real scenarios often involve bounded diffusion layers, as in thin-layer cells or coated samples. In such cases, the impedance at lower frequencies deviates from the “infinite” Warburg equation and is better described by the “finite” Warburg impedance, given by Equation 6.

In the “finite” Warburg equation:

– Z represents the impedance.

– δ stands for the Nernst diffusion layer thickness.

– D is an average value of the diffusion coefficients of the diffusing species.

– ω denotes the radial frequency of the perturbation.

At high frequencies (ω →∞), or when the diffusion layer thickness is effectively infinite (δ → ∞), the term (tanh δ (jw/ D)1/2 approaches 1. In these conditions, the “finite” Warburg equation simplifies towards the “infinite” Warburg impedance.

The admittance parameter (Y0) is defined as (Y0= 12) offering an alternative representation of the Warburg impedance. It’s essentially the reciprocal of the product of the Warburg coefficient and the square root of 2.

Both the “infinite” and “finite” Warburg impedance equations help explain the behavior of electrochemical systems under different diffusion limitations, accounting for the thickness of the diffusion layer and its impact on impedance at varying frequencies.

**Table 1: Circuit elements used in the models.**

Equivalent element |
Impedance |

R | R |

C | 1C |

L | ωL |

Q (Constant phase element) | 1(jω)Y0 |

W (Warburg finite) | 1/Y0√(jω) |

O (Warburg infinite) | σ(ω)-1/2 (1-J) tanh δ (jD)1/2 |

Electrochemical Impedance Spectroscopy is a versatile technique providing a comprehensive view of an electrochemical system’s behavior across various frequencies. It offers valuable insights into reaction kinetics, system impedance, and interface properties, serving as a fundamental tool in electrochemistry for studying electrochemical interfaces and processes.