Electrochemistry MAE  212. Dr. Marc Madou, UCI, Winter 2014 Class X Electrochemical Impedance Analysis (EIS). Table of Content. Introduction Advantages and Disadvantages Impedance Concept and Representation of Complex Impedance Values Nyquist plot or ColeCole plot Bode plot
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Dr. Marc Madou, UCI, Winter 2014
Class X Electrochemical Impedance Analysis (EIS)
Z(t)=E(t)/I(t)
Multiplier:
Vx(t)sin(t) &
Vx(t)cos(t)
Integrator: integrates multiplied signals
Display result:
a + jb = Vsign/Vref
Impedance:
Zsample = Rm (a + jb)
But be aware of the input impedance of the FRA!
Vpwr.amp = A k Vk
A= amplification
Vwork – Vref =
Vpol. + V3 + V4
Currentvoltage converter provides virtual ground for Workelectrode.
Source of inductive effects
General schematic
E(t) = E0 exp(it)
E(t) = E0cos(t), =2f
I(t) = I0 cos(t)
Or, if one writes in complex notation:
E(t) = E0 exp(it)
I(t) = I0 exp(it  i)
Z(t) = Z0 exp(i) = Z0 (cos + isin)
E/I response for a resistor (=0)
E/I response for a capacitor (=90)
E/I response for an inductor (=90)
Impedance ‘resistance’
Admittance ‘conductance’:
Representation of impedance value, Z = a +jb, in the complex plane (see also http://math.tutorvista.com/numbersystem/absolutevalueofacomplexnumber.html
What is the impedance of an RC circuit?
Admittance?
Semi
circle
‘time constant’:
= RC
R Values
C
Data Presentation:Nyquist Plot with Impedance Vectorfmax = 1/(6.3x3109x105)=530 Hz
R = 100 k
C = 3 nF
518 Hz
1 MHz
1 Hz
<< :
>> :
This is best observed in a socalled Bode plot
log(Zre), log(Zim) vs. log(f), or
logZ and phase vs. log(f)
Very few electrochemical cells can be modeled using a single equivalent circuit element. Instead, EIS models usually consist of a number of elements in a network. Both serial and parallel combinations of elements occur.
Impedances in Series:
Impedances in Parallel
R1
R2
C1
C2
= Values
Z
R
F
=
0
o
1
=
Z
w
C
F
=

90
o
Review of Circuits ElementsI
Resistance:
E
E
Capacitance:
I
Electrochemical Impedance Spectroscopy Values
Equivalent Circuit of a Cell: Rs and CdA Resistance and capacitance
in series
In electrochemical cell:
R=Rs: solution resistance
C=Cd: double layer capacitance
f is low:
f is high:
Principle of the
Electric DoubleLayer: Here C electrodes
Deviation from ‘ideal’ dispersion:
Constant Phase Element (CPE),
(symbol: Q )
n = 1, ½,
0, 1, ?
How to explain this nonideal behaviour?
1980’s: ‘Fractal behaviour’ (Le Mehaut)
= fractal dimensionality
i.e.: ‘What is the length of the coast line of England?’
Depends on the size of the measuring stick!
Self similarity
Mixed kinetic and diffusion control
C
dl or CPE
R
W
with 0n1
RP
ZW
In the forward reaction in the first equation, electrons enter the metal and metal ions diffuse into the electrolyte. Charge is being transferred.
This charge transfer reaction has a certain speed. The speed depends on the kind of reaction, the temperature, the concentration of the reaction products and the potential.
The general relation between the potential and the current holds:
io = exchange current density
Co = concentration of oxidant at the electrode surface
Co* = concentration of oxidant in the bulk
CR = concentration of reductant at the electrode surface
F = Faradays constant
T = temperature
R = gas constant
a = reaction order
n = number of electrons involvedh= overpotential ( E  E0 )
The overpotential, h, measures the degree of polarization. It is the electrode potential minus the equilibrium potential for the reaction.
When the concentration in the bulk is the same as at the electrode surface, Co=Co* and CR=CR*. This simplifies the last equation into:
This equation is called the ButlerVolmer equation. It is applicable when the polarization depends only on the charge transfer kinetics.
Stirring will minimize diffusion effects and keep the assumptions of Co=Co* and CR=CR* valid.
When the overpotential, h, is very small and the electrochemical system is at equilibrium, the expression for the charge transfer resistance changes into:
From this equation the exchange current i0 density can be calculated when Rct is known (see Rct in next figure).
A resistance and capacitance in parallel (Randles circuit)
Z=Rs at high frequency
Z=Rct+Rs at low frequency
When there are two simple, kinetically controlled reactions occurring, the potential of the cell is again related to the current by the following (known as the ButlerVolmer equation).
I is the measured cell current in amps,Icorr is the corrosion current in amps,Eoc is the open circuit potential in volts,a is the anodic Beta coefficient in volts/decade,c is the cathodic Beta coefficient in volts/decade
If we apply a small signal approximation (EEoc is small) to the buler Volmer equation, we get the following:
which introduces a new parameter, Rp, the polarization resistance.
If the Tafel constants i are known, you can calculate the Icorr from Rp. The Icorr in turn can be used to calculate a corrosion rate. The Rpparameter comes again from the Nyquist plot.
Diffusion can create an impedance known as the Warburg impedance. 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.
The equation for the "infinite" Warburg impedance
On a Nyquist plot the infinite Warburg impedance appears as a diagonal line with a slope of 0.5. On a Bode plot, the Warburg impedance exhibits a phase shift of 45°.
In the above equation, s is the Warburg coefficient defined as:
= radial frequency
DO = diffusion coefficient of the oxidant
DR = diffusion coefficient of the reductant
A = surface area of the electrode
n = number of electrons transferred
C* = bulk concentration of the diffusing species (moles/cm3)
In solution: Values
Equivalent Circuit of a Cell: Warburg ImpedanceDefine impedance in Laplace space!
Take the Laplace variable, p, complex:
p = s + j. Steady state: s 0,
which yields the impedance:
A capacitor is formed when two conducting plates are separated by a nonconducting media, called the dielectric. The value of the capacitance depends on the size of the plates, the distance between the plates and the properties of the dielectric. The relationship is:
With,
o = electrical permittivity
r = relative electrical permittivity
A = surface of one plate
d = distances between two plates
Whereas the electrical permittivity is a physical constant, the relative electrical permittivity depends on the material. Some useful r values are given in the table:
Notice the large difference between the electrical permittivity of water and that of an organic coating. The capacitance of a coated substrate changes as it absorbs water. EIS can be used to measure that change
A metal covered with an undamaged coating generally has a very high impedance. The equivalent circuit for such a situation is in the Figure:
C
R
The model includes a resistor (due primarily to the electrolyte) and the coating capacitance in series.
A Nyquist plot for this model is shown in the Figure. In making this plot, the following values were assigned:
R = 500 (a bit high but realistic for a poorly conductive solution)C = 200 pF (realistic for a 1 cm2 sample, a 25 µm coating, and r = 6 )fi = 0.1 Hz (lowest scan frequency  a bit higher than typical)
ff = 100 kHz (highest scan frequency)
The value of the capacitance cannot be determined from the Nyquist plot. It can be determined by a curve fit or from an examination of the data points. Notice that the intercept of the curve with the real axis gives an estimate of the solution resistance.
The highest impedance on this graph is close to 1010. This is close to the limit of measurement of most EIS systems
The same data from the previous page shown in a Bode plot below. Notice that the capacitance can be estimated from the graph but the solution resistance value does not appear on the chart. Even at 100 kHz, the impedance of the coating is higher than the solution resistance
Classification of types of capacitances
source approximate value
geometric 220 pF (cm1)
grain boundaries 110 nF (cm1)
double layer / space charge 0.110 F/cm2
surface charge /”adsorbed species” 0.2 mF/cm2
(closed) pores 1100 F/cm3
“pseudo capacitances”
“stoichiometry” changes large !!!!
Below we show some common equivalent circuits models. Equations for both the admittance and impedance are given for each element.
By using the various ColeCole plots we can calculate values of the elements of the equivalent circuit for any applied bias voltage
By doing this over a range of bias voltages, we can obtain:
the field distribution in the layers of the device (potential divider) and the relative widths of the layers, since C ~ 1/d
KramersKronigrelations
Real and imaginary parts are linked through the KK transforms:
Response only due to input signal
Response scales linearly with input signal
State of system may not change during measurement
not a singularity! Coating
Note: Putting ‘KK’ in practiceRelations,
Real imaginary:
Imaginary real: