**Electrochemical Impedance Spectroscopy**

Artigo de revisão sobre espectroscopia de impedância eletroquímica.

(Parte **1** de 12)

Electrochemical Impedance Spectroscopy and its Applications

Andrzej Lasia

Département de chimie, Université de Sherbrooke, Sherbrooke Québec, J1K 2R1 alasia@courrier.usherb.ca

A. Lasia, Electrochemical Impedance Spectroscopy and Its Applications, Modern Aspects of Electrochemistry, B. E. Conway, J. Bockris, and R.E. White, Edts., Kluwer Academic/Plenum Publishers, New York, 1999, Vol. 32, p. 143-248.

I. Introduction | 4 |

1. Response of Electrical Circuits | 4 |

(i) Arbitrary Input Signal | 4 |

(i) Alternating Voltage (av) Input Signal | 6 |

2. Impedance of Electrical Circuits | 7 |

(i) Series R-C Circuit | 8 |

(i) Parallel R-C Circuit | 8 |

(i) Series: Rs + Parallel R-C Circuit | 9 |

3. Interpretation of the Complex Plane and Bode Plots | 9 |

I. Im pedance measurements | 10 |

1. Ac Bridges | 10 |

2. Lissajous Curves | 1 |

3. Phase Sensitive Detection (PSD) | 1 |

4. Frequency Response Analyzers | 12 |

5. Fast Fourier Transform (FFT) | 13 |

(i) Pulse Perturbation | 14 |

(i) Noise Perturbation | 14 |

(i) Sum of Sine Waves | 14 |

I. Impedance of faradaic reactions in the presence of diffusion | 15 |

1. The Ideally Polarizable Electrode | 15 |

2. Semi-Infinite Linear Diffusion | 15 |

3. Spherical Diffusion | 20 |

4. Cylindrical Electrodes | 2 |

5. Disk Electrodes | 23 |

6. Finite-Length Diffusion | 23 |

(i) Transm issive Boundary | 24 |

(i) Reflective Boundary | 25 |

Kinetic Parameters | 26 |

(i) Randles’ Analysis64,65,67 | 26 |

(i) De Levie-Husovsky Analysis | 27 |

(i) Analysis of cot ϕ | 27 |

(iv) Sluyter' s Analysis | 28 |

IV. Impedance of A faradaic reaction involving adsorption of reacting species | 29 |

1. Faradaic Reaction Involving One Adsorbed Species | 29 |

2. Impedance Plots in the Case of One Adsorbed Species | 31 |

3. Faradaic Impedance in the Case Involving Two Adsorbed Species | 34 |

4. Impedance Plots in the Case of Two Adsorbed Species | 36 |

5. Faradaic Impedance for the Process Involving Three or More Adsorbed Species | 36 |

Table of content 7. Analysis of Impedance Data in the Case of Semi-Infinite Diffusion: Determination of the V. Impedance of solid electrodes................................................................................................37

1. Frequency Dispersion and Electrode Roughness | 37 |

2. Constant Phase Element | 37 |

3. Fractal Model | 40 |

4. Porous Electrode Model | 42 |

(i) Porous Electrodes in the Absence of Internal Diffusion | 42 |

(a) De Levie’s treatment | 43 |

(b) Rigorous treatment | 45 |

(i) Porous Electrodes in the Presence of Axial Diffusion | 46 |

(i) Other Pore Geometries | 49 |

5. Generalized Warburg Element | 49 |

VI. Conditions for "good" impedances | 50 |

1. Linearity, Causality, Stability, Finiteness | 50 |

2. Kram ers-Kronig Transforms | 51 |

3. Non-Stationary Impedances | 53 |

VII. Modeling of experimental data | 54 |

1. Selection of the Model | 54 |

2. CNLS Approximations | 56 |

(i) CNLS Method | 56 |

(i) Statistical Weights | 57 |

(i) AC Modeling Programs | 58 |

VIII. Instrumental limitations | 58 |

Electrochemical Impedance Spectroscopy (EIS) or ac impedance methods have seen tremendous increase in popularity in recent years. Initially applied to the determination of the double-layer capacitance1-4 and in ac polarography,5-7 they are now applied to the characterization of electrode processes and complex interfaces. EIS studies the system response to the application of a periodic small amplitude ac signal. These measurements are carried out at different ac frequencies and, thus, the name impedance spectroscopy was later adopted. Analysis of the system response contains information about the interface, its structure and reactions taking place there. EIS is now described in the general books on electrochemistry,8-17 specific books18,19 on EIS, and there are also numerous articles and reviews.6,20-31 It became very popular in the research and applied chemistry. The Chemical Abstract database shows ~1,500 citations per year of the term "impedance" since 1993 and ~1,200 in earlier years and ~500 citations per year of "electrochemical impedance". Although the term "impedance" may include also nonelectrochemical measurements and "electrochemical impedance" may not include all the electrochemical studies, the popularity of this technique cannot be denied.

However, EIS is a very sensitive technique and it must be used with great care. Besides, it is not always well understood. This may be connected with the fact that existing reviews on EIS are very often difficult to understand by non-specialists and, frequently, they do not show the complete mathematical developments of equations connecting the impedance with the physico-chemical parameters. It should be stressed that EIS cannot give all the answers. It is a complementary technique and other methods must also be used to elucidate the interfacial processes.

The purpose of this review is to fill this gap by presenting a modern and relatively complete review of the subject of electrochemical impedance spectroscopy, containing mathematical development of the fundamental equations.

1. Response of Electrical Circuits (i) Arbitrary Input Signal

Application of an electrical perturbation (current, potential) to an electrical circuit causes the appearance of a response. In this chapter, the system response to an arbitrary perturbation and, later, to an ac signal, will be presented. Knowledge of the Laplace transform technique is assumed, but the reader may consult numerous books on the subject.

First, let us consider application of an arbitrary (but known) potential E(t) to a resistance

R. The current i(t) is given as: i(t) = E(t)/R. When the same potential is applied to the series connection of the resistance R and capacitance C, the total potential difference is a sum of potential drops on each element. Taking into account that for a capacitance E(t) = Q(t) /C, where Q is the charge stored in a capacitor, the following equation is obtained:

Et i t R Qt

C it R C

This equation may be solved using either Laplace transform or differentiation techniques. 32-34 Differentiation gives:

di tdt it RC R dE tdt which may be solved for known E(t) using standard methods for differential equations.

The Laplace transform is an integral transform in which a function of time f(t) is transformed into a new function of a parameter s called frequency, fs() or F(s), according to:

(3)

The Laplace transform is often used in solution of differential and integral equations. In general, the parameter s may be complex, s = ν +jω, where j = −1, but in this chapter only the real transform will be considered, i.e. s = v. Direct application of the Laplace transform to eqn. (1), taking into account that L (itdtt

which leads to:

The ratio of the Laplace transforms of potential and current, E(s)/i(s) is expressed in the units of resistance, Ω, and is called impedance, Z(s). In this case:

Zs Esis R sC () ()()

The inverse of impedance is called admittance, Y(s) = 1/Z(s). They are transfer functions which transform one signal, e.g. applied voltage, into another, e.g. current. Both are called immittances. Some other transfer functions are discussed in refs. 18, 35 and 36. It should be noticed that the impedance of a series connection of a resistance and capacitance, eqn. (6), is a sum of the contributions of these two elements: resistance, R, and capacitance, 1/sC.

For the series connection of a resistance, R, and inductance, L, the total potential difference consists of the potential drop on both elements:

Et i t R L di tdt ()()()=+ (7)

In both cases considered above the system impedance consists of the sum of two terms, corresponding to two elements: resistance and capacitance or inductance.

In general, one can write contributions to the total impedance corresponding to the resistance as R, the capacitance as 1/sC and the inductance as sL. Addition of impedances is analogous to the addition of resistances. Knowledge of the system impedance allows for an easy solution of the problem.

For example, when a constant voltage, E0, is applied at time zero to a series connection of R and C, the current is described by eqn. (5). Taking into account that the Laplace transform of a constant L (E0) = E0 /s, one gets:

is E sR sC E

(9)

Inverse transform of (9) gives the current relaxation versus time:

it E

The result obtained shows that after the application of the potential step, current initially equals

E0/R and it decreases to zero as the capacitance is charged to the potential difference E0. Similarly, application of the potential step to a series connection of R and L produces response given by eqn. (8) which, after substitution of E(s) = E0 /s, gives:

is E

sR sL E

Ls s R L E Rs s R L()

Inverse transform gives the time dependence of the current:

The current starts at zero as the inductance constitutes infinite resistance at t = 0 and it increases to E0/R as the effect of inductance becomes negligible in the steady-state condition. In a similar way other problems of transient system response may be solved. More complex examples are presented, e.g., in refs. 3-34. It should be added that an arbitrary signal may be applied to the system and if the Laplace transforms of the potential and current are determined, e.g. by numerical transform calculations, the system impedance is determined. In the Laplace space the equations (e.g. eqns. (9) and (1)) are much simpler than those in the time space (e.g. eqns. (10) and (12)) and analysis in the frequency space s allows for the determination of the system parameters. This analysis is especially important when an ideal potential step cannot be applied to the system because of the band-width limitations of the potentiostat.37 In this case it is sufficient to know i(t) and the real value of the potential applied to the electrodes by the potentiostat, E(t), which allows numerical Laplace transformation to be carried out and the system impedance obtained.

In the cases involving more time constants, i.e. more than one capacitance or inductance in the circuit, the differential equations describing the system are of the second or higher order and the impedances obtained are the second or higher order functions of s.

(i) Alternating Voltage (av) Input Signal In the EIS we are interested in the system response to the application of a sinusoidal signal, e.g.: E = E0 sin(ωt), where E0 is the signal amplitude, ω = 2πf is the angular frequency, and f is the av signal frequency. This problem may be solved in different ways. First, let us consider application of an av signal to a series R-C connection. Taking into account that the

Laplace transform of the sine function L [sin(ωt)] = ω/(s2 + ω2)], use of eqn. (5) gives:

is E

s R sC E

Distribution into simple fractions leads to:

is E

it E

R C t RC t RC tR C()

The third term in eqn. (15) corresponds to a transitory response observed just after application of the av signal and it decreases quickly to zero. The steady-state equation may be rearranged into a simpler form:

t RC ωωω (16) and by introducing tan ϕ = 1/ωRC the following form is found:

(Parte **1** de 12)