Transcript Techniques based on concepts of impedance

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.
• 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.
Electrochemical Impedance Spectroscopy
FRA: Frequency Response
I
Analysis
I0
I0 +  I sin ( t + 
E0
Potentiostatic or galvanostatic
measurements
E
E0 +  E sin t
Review of ac circuits
• A purely sinusoidal voltage can be expressed as
e  E sin t
• Where ω is the angular frequency, which is 2π
times the conventional frequency in hertz.
• 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.
• A pure capacitance C,
E

•
i
sin(t  )
XC
2
• Where Xc is the capacitive reactance, 1/ωC
• A comparison of R and Xc shows that Xc must carry
dimensions of resistance, but the magnitude of Xc
falls with increasing frequency.
Electrochemical Impedance Spectroscopy
Resistance:
Capacitance:
Z R
Z 
1
 C
I
E
E
I
F 0
o
F  -90
o
Review of ac circuits
• Components along the ordinate are
assigned as imaginary and along the
abscissa are real, thus, we handle these
parameters mathematically as “real” or
“imaginary”.
E  - jX C I
• A voltage E is applied across R and C
E  ER  EC  I (R - jX C )  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  ( R 2  X C 2 )1/ 2
XC
1
tan F 

R  RC
• The impedance is a kind of generalized resistance. The
phase angle expresses the balance between capacitive
and resistance components in the series circuit. For a
pure resistance, φ=0; for a pure capacitance, φ=π/2; and
for mixtures, intermediate phase angles are observed.
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.
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.
• 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.
Electrochemical Impedance Spectroscopy
A Resistance and capacitance
in series
1
f is low: Z 
 C
f is high: Z  R
F  90
F0
In electrochemical cell:
R=Rs: solution resistance
C=Cd: double layer capacitance


Bode
Nyquist
9
10
|Z|, W
Zj, W
-600
-400
-200
0
500.00
Zr, W
10
10
10
8
-90
6
-80
Phase
-800x10
4
-60
2
10
-70
1
10
2
10
3
10
4
10
Frequency, Hz
5
10
6
10
1
10
2
10
3
10
4
10
Frequency, Hz
5
10
6
Electrochemical Impedance Spectroscopy
A resistance and capacitance in parallel (Randles circuit)
Z=Rs at high frequency
Z=Rct+Rs at low frequency
Nyquist
Bode
-300
2
-50
Zj, W
-200
-150
-40
|Z|, W
Phase
-250
-30
-20
0
10
-50
0
10
2
10
4
Frequency, Hz
0
0
100
200
Zr, W
300
8
6
4
-10
-100
100
10
6
2
0
10
2
10
4
10
Frequency, Hz
6
10
Electrochemical Impedance Spectroscopy
Mixed kinetic Cand diffusion control
dl or CPE
Z  -1 / (Q)
RW
RP
with 0n1
ZW
Bode
Nyquist
-200
-40
100
-100
7
6
5
Phase
|Z|, W
Zj, W
-150
-30
-20
4
-50
3
0
2
0
50
100
Zr, W
150
200
-10
0
0
10
2
10
Frequency, Hz
4
10
0
10
2
10
Frequency, Hz
4
10
n
Electrochemical Impedance Spectroscopy
Data Analysis:
•Abstract the cell into an equivalent circuit
model
•Before starting to fit, get good initial guess
values
Use „find circle“ option
Use linear regression to evaluate a Warburg
impedance
•Take care of:
Non uniqueness of equivalent circuit models
Weighting the data
b
a
30
3
3
10
1
20
I/mA
I/mA
1
0
10
-10
0
0.2
0.4
0.6
E/V
0.8
0.2
0.4
0.6
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
0.8
30
20
5
1
15
Ip,a/mA
I/mA
15
0
10
-15
0.2
0.4
0.6
E/V
0.8
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
36
5
30
1
Il/mA
Il/A
30
15
24
18
0
0.3
0.6
E/V
0.9
6
8
10
1/2
12
-1/2
 /(s )
Left: Voltammetric curves of 1 mmol/L ferrocene on the
GC eletrode with different rotation rate, rotation rate
(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
Z"/kW
24
16
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.
25
20
Z"/kW
15
10
5
0
0
5
10
15
20
25
Z'/kW
Left: Nyquist plots of 1 mmol/L ferrocene on the rotating
GC electrode and its fitting results (solid line), E=0.50 V.
Right: the corresponding equivalent circuit.
Diffusion coefficient and standard
heterogeneous rate constant
1
Cd 
 B Rct
DC
AD
RT  1
1 

 1/ 2 *  1/ 2 * 
nFA 2  DO CO DR C R 
RT
Rct 
nFi0

Rct - AD
2Cd