( electrochemical impedance spectroscopy, EIS) Chapter 6 : Alternative current method ( electrochemical impedance spectroscopy, EIS)

6.1 basic consideration 1. Probing electrochemical system circuit analysis : black box grey box for electrochemical system: Rc t RS Cd t R.E W.E electric behavior of elements : resistor, capacitor, etc.

Equivalent circuit and circuit description code: CDC

Sinusoidal Current Response in a Linear System The excitation signal is a function of time E0 is the amplitude of the signal, and  is the radial frequency In a linear system, the response signal, It, is shifted in phase () and has an amplitude of I0

An expression analogous to Ohm's Law allows us to calculate the impedance of the system as: 阻抗 (Impedance) (Z) 导纳 (Admittance) (Y)

EIS models usually consist of a number of elements in a network EIS models usually consist of a number of elements in a network. Both serial and parallel combinations of elements occur. Impedances in Series    Z1 Z2 Z3 For linear impedance elements in series, the equivalent impedance is

Impedances in Parallel  Z1 Z2 Z3 For linear impedance elements in parallel, the equivalent impedance is

If we plot the sinusoidal signal on the X-axis of a graph and the sinusoidal response signal I(t) on the Y-axis, an oval is plotted. Analysis of oval figures on oscilloscope screens was the method of impedance measurement prior to the lock-in amplifiers and frequency response analyzers

2. Display of impedance Using Eulers relationship it is possible to express the impedance as a complex function. The potential is described as, and the current response as The impedance is then represented as a complex number,

The expression for Z() is composed of a real part (Z’) and an imaginary part (Z’’). If the real part is plotted on the Z’ axis and the imaginary part on the -Z’’ axis of a chart, Nyquist plot is gotten.

Nyquist Plots A Nyquist plot is made up of a series of vectors representing the total magnitude of the resistance and capacitance components Phase angle Non Resistive Component

Bode impedance plot Impedance Rct Solution resistance Frequency →

6.2 Electrochemical elements: 1) Electrolyte resistance (uncompensated resistance) (Rs, RU) 2) Double layer capacitance (Cdl) 3) Coating capacitance (Cc) 4) Warburg impedance (related to diffusion) (W) 5) Polarization resistance/Charge transfer resistance (Rct i0) 6) Constant phase element (Q) 7) Virtual inductor (L)

1) Electrolyte resistance (uncompensated resistance) (Rs, RU) 2) Double layer capacitance (Cdl) 3) Coating capacitance (Cc) Conversion film, passivation film, polymeric coating, etc. Typical Relative Electrical Permittivity vacuum 1 water 80.1 ( 20 ℃ ) organic coating 4 - 8

4) Warburg impedance: related to diffusion At x = 0

If product is insoluble: depends on ω. When ω→∞

韦伯格系数（Warburg factor ）

If product is soluble:

This form of the Warburg impedance is only valid if the diffusion layer has an infinite thickness. Quite often this is not the case. If the diffusion layer is bounded, the impedance at lower frequencies no longer obeys the equation above. Instead, we get the form:  Is the thickness of the diffusion layer

←Frequency This impedance depends on the frequency of the potential perturbation. At high frequencies the Warburg impedance is small since diffusing reactants don't have to move very far. At low frequencies the reactants have to diffuse farther, thereby increasing the Warburg impedance. On a Nyquist plot the infinite Warburg impedance appears as a diagonal line with a slope of 0.5.

5) Charge transfer resistance (Rct) Without concentration overpotential At small overpotential At higher overpotential For medium overpotential and ==0.5

With concentration overpotential

6) Constant Phase Element Capacitors in EIS experiments often do not behave ideally. Instead, they act like a constant phase element (CPE) . For an ideal capacitor, the constant A = 1/C (the inverse of the capacitance) and the exponent  = 1. For a constant phase element, the exponent  is less than one and of no definite physical meaning.

7) Virtual Inductor ZC = jL ZC=0 ZC = L The impedance of an electrochemical cell can also appear to be inductive. Some authors have ascribed inductive behavior to adsorbed reactants. Both the adsorption process and the electrochemical reaction are potential dependent. The net result of these dependencies can be an inductive phase shift in the cell current . Inductive behavior can also result from nonhomogeneous current distribution, which lead inductance and potentiostat non-idealities. In these cases, it represents an error in the EIS measurement.

Common Equivalent Circuit Models 1/  Y0( j)            Y0( j)       Q (CPE) O (finite Warburg) 1/Y0( j)1/2      Y0( j)1/2            W (infinite Warburg) jL     1/  jL          L    1/ jC         jC    C R       1/R       R Impedance Admittance Equivalent element The dependent variables are R, C, L, Yo, B and a.

6.3 Simplification of EC For electrode with metal current collector，RCE→0，RWE→0 Z = 1/ jC        Compare with CWE and CCE, CW-C is very small. Therefore, the above circuit can be simplified as

How can we further simplify this circuit? Z1 Cdl ,2 RL Cdl ,1 How can we further simplify this circuit? 1)When using electrode with large effective area and exchange current . Cdl very large 1/Cd very small RL used for measurement of conductivity of solution

2) When using reference electrode : Rct Z1 Cdl ,2 RL Cdl ,1 2) When using reference electrode : RS Rct W.E 1) No rxn , Rct  , RL Cdl Cdl if 1/Cdl >> 0 ideal polarization electrode

6.4 impedance measurement variatory :  106 ~ 10-3 Hz single generator :  from 105 ~ 10-3 Hz 5 ~ 10 point /decade v0 = 5 mV for high impedance system :10 mV lock  in amplifier : measure : frequency  respond analyzer : z =z  zj z f z z Nyquit

EIS (Summary) We start here at the high frequency

6.5 Impedance characteristics of processes 1) Ideally Polarizable Electrode An ideally polarizable electrode behaves as an ideal capacitor because there is no charge transfer across the solution-electrode boundary. Circuit code: RsCdl

Rs

RS Cdl 2) For (RC) Randles Cell

Capacitive impedance semicircle

3) EC with and without diffusion WO WR Rct Cdl Rs This circuit models a cell with polarization due to a combination of kinetic and diffusion processes At low frequency

the Warburg Impedance appears as a straight line with a slope of 45°

At higher frequency When i0, Rct  0, no circle appears.

The whole spectrum

4) For coated metal system there are two well defined time constants in this plot

6) RL and (RL) RL R (RL)

0<n<1 6) RQ and (RQ) Q independence of  (RQ) RQ

7) Uniqueness of Models This spectrum can be modeled by any of the equivalent circuits You cannot assume that an equivalent circuit that produces a good fit to a data set represents an accurate physical model of the cell

elements CoPc FePc FeCoPc2 L/ H·cm-2 7e-7 5.788e-7 7.851e-7 Rs/ ·cm-2 1.677 2.024 1.598 Rct/ ·cm-2 3.05 1.966 0.6907 Rc1/ ·cm2 0.6397 0.9987 0.4835 Rc2/ ·cm-2 2.238 2.204 0.4143 Rtotal/ ·cm-2 7.6047 7.1927 3.1865