• The elements of equivalent circuit of a cell: double-layer capacitance Cd, faradaic impedance Zf, solution resistance Rs, charge transfer resistance Rct, Warburg impedance Zw.
(r/min): (1) 300, (2) 600, (3) 900, (4) 1200, (5) 1500; scan
rate: 5 mV/s; Right: plot of limiting diffusion current (Il) square root of angular velocity (ω1/2) for ferrocene
E - jXC I
• A voltage E is applied across R and C
E ER EC I (R - jXC ) IZ
• Where Z=R-jXc, called the impedance.
Review of ac circuits
• The magnitude of Z and phase angle are given by the following, respectively Z (R2 X C2 )1/ 2
Mixed kinetic and diffusion control Cdl or CPE RW
Z -1 / (Q)n
RP
ZW
with 0n1
Zj, W |Z|, W Phase
Nyquist
-200
-150
-100
-50
0 0 50 100 150 200 Zr, W
Bode
100
7 6 5 4 3
Zj, W |Z|, W Phase
-800x109 -600 -400 -200 0
Nyquist
500.00 Zr, W
Bode
8
10
-90
106
-80
4
10
2
10
1
2
3
4
5
6
10 10 10 10 10 10
Frequency, Hz
-70
-60
1
2
3
4
5
6
10 10 10 10 10 10
Frequency, Hz
I/mA Ip,a/mA
30 15 0 -15
0.2
5 1
0.4
0.6
0.8
E/V
20
15
10
5
0.1
0.2
0.3
0.4
n1/2/(V·s-1）
Cyclic voltammograms of 1 mmol/L ferrocene on the GC
electrode, Plot of oxidation peak current (Ip,a) vs. square root of scan rate (υ1/2) for ferrocene
• Another approach is to perturb the cell with an alternating signal of small magnitude and observe the way in which system follows the perturbation at steady state.
✓Use „find circle“ option ✓Use linear regression to evaluate a Warburg impedance
•Take care of:
✓Non uniqueness of equivalent circuit models ✓Weighting the data
Electrochemical Impedance Spectroscopy
A Resistance and capacitance in series
f is low:
1 Z
C
f is high: Z R
F 90o F 0o
In electrochemical cell:
R=Rs: solution resistance C=Cd: double layer capacitance
Review of ac circuits
• A purely sinusoidal voltage can be expressed as
e E sint
• Where ω is the angular frequency, which is 2π times the conventional frequency in hertz.
I/mA
I/mA
a
3
10
1
0
b
3
30
1
20
10
-10
0
0.2
0.4
0.6
0.8
E/V
0.2
0.4
0.6
0.8
E/V
Cyclic voltammograms(a) and differential pulse voltammograms(b) of different concentrations of ferrocene on the GC electrode, C(mmol/L): (1 ) 0.2, (2) 0.5, (3) 1; scan rate: 100 mV/s
Equivalent circuit of a cell
• In a general sense, we ought to be able to represent its performance by an equivalent circuit of resistors and capacitors under a given excitation.
Electrochemical Impedance Spectroscopy
FRA: Frequency Response
Analysis
I
I0 I0 + I sin ( t +
E0
Potentiostatic or galvanostatic
E
measurements
E0 + E sin t
Techniques based on concepts of
impedance
• We have discussed ways of studying electrode reactions through large perturbations on the system, for example, potential sweeps, potential steps, or current steps, the electrode is generally driven to a condition far from equilibrium and the response is observed, which is usually a transient signal.
Review of ac circuits
• For impedances in parallel, the inverse of the overall impedance is the sum of the reciprocals of the individual vectors. Sometimes it is advantageous to analyze ac circuits in terms of the admittance, Y, which is the inverse impedance 1/Z.
• The current lags the voltage, it can be expressed generally as
i I sin(t )
• Where φ is a phase angle.
Review of ac circuits
• A pure resistance R, E=IR, where the phase is zero.
24
16
Z"/kW
8
0
0
20
40
60
Z'/kW
Left: Nyquist plots of 1 mmol/L ferrocene at different potentials on the rotating GC electrode and its fitting results (solid line), E (V):■0.40,●0.45,▲0.50; rotation rate=900 r/min; Right: the corresponding equivalent circuit.
-300 -250 -200 -150 -100
-50 0 0
Nyquist
100
200
300
Zr, W
Bode
-50
-40
-30
-20
-10
0
0
2
4
6
10 10 10 10
Frequency, Hz
2
100
8 6
4
2
0
2
4
6
10 10 10 10
Frequency, Hz
Electrochemical Impedance Spectroscopy
Electrochemical Impedance Spectroscopy
A resistance and capacitance in parallel (Randles circuit)