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Slide 1
Dr. Marc Madou, UCI, Winter 2012 Class X Electrochemical
Impedance Analysis (EIS) Electrochemistry MAE-295
Slide 2
Table of Content 2 Introduction Advantages and Disadvantages
Impedance Concept and Representation of Complex Impedance Values
Nyquist plot or Cole-Cole plot Bode plot Summary Nyquist and Bode
Plots Review of Circuits Elements Equivalent Circuit of a Cell Back
to Electrochemistry Summary
Slide 3
Introduction In EIS an electrochemical system is perturbed with
an alternating current or voltage signal of small magnitude and one
observes the way in which the system follows the perturbation at
steady state. EIS measures the impedance of a circuit to an applied
voltage: Z(t)=E(t)/I(t) When E (or I) is applied as a sinusoidal
function in a linear system, the I (or V) response can be
represented by a sum of sinusoidal functions with phase shifts. If
an equivalent circuit for the system being probed can be
constructed, then the resistance or capacitance values for each
circuit element can be backed out from Z. 3
Slide 4
Introduction Electrical circuit theory distinguishes between
linear and non-linear systems (circuits). Impedance analysis of
linear circuits is much easier than analysis of non-linear ones. A
linear system... is one that possesses the important property of
superposition: If the input consists of the weighted sum of several
signals, then the output is simply the superposition, that is, the
weighted sum, of the responses of the system to each of the
signals. Mathematically, let y 1 (t) be the response of a
continuous time system to x 1 (t) ant let y 2 (t) be the output
corresponding to the input x 2 (t). Then the system is linear if:
1)The response to x 1 (t) + x 2 (t) is y 1 (t) + y 2 (t) 2) The
response to ax 1 (t) is ay 1 (t)... 4
Slide 5
For a potentiostated electrochemical cell, the input is the
potential and the output is the current (for a galvanostated cell
it is the other way around). Most electrochemical cells are not
linear! Doubling the voltage will not necessarily double the
current. However, electrochemical systems can be pseudo-linear.
When you look at a small enough portion of a cell's current versus
voltage curve, it seems to be linear. In normal EIS practice, a
small (1 to 10 mV) AC signal is applied to the cell. The signal is
small enough to confine you to a pseudo-linear segment of the
cell's current versus voltage curve. You do not measure the cell's
nonlinear response to the DC potential because in EIS you only
measure the cell current at the excitation frequency. Introduction
5
Slide 6
Measure Z( ,V bias ) = V( ) / I( ) 6
Slide 7
Introduction Let us assume we have an electrical element to
which we apply an electric field E(t) and get the response I(t),
then we can disturb this system at a certain field E with a small
perturbation dE and we will get at the current I a small response
perturbation dI. In the first approximation, as the perturbation dE
is small, the response dI will be a linear response as well. If we
plot the applied sinusoidal signal on the X-axis of a graph and the
sinusoidal response signal I(t) on the Y-axis, an oval is plotted.
This oval is known as a "Lissajous figure". Analysis of Lissajous
figures on oscilloscope screens was the accepted method of
impedance measurement prior to the availability of lock-in
amplifiers (LIAs) and frequency response analyzers (FRAs). 7
Slide 8
Introduction Multiplier: V x ( t) sin( t) & V x ( t) cos(
t) Integrator: integrates multiplied signals Display result: a + jb
= V sign /V ref Impedance: Z sample = R m (a + jb) But be aware of
the input impedance of the FRA! 8
Slide 9
Introduction General schematic V pwr.amp = A k V k A=
amplification V work V ref = V pol. + V 3 + V 4 Current-voltage
converter provides virtual ground for Work-electrode. Source of
inductive effects 9
Slide 10
Introduction 10
Slide 11
Introduction FRA: Frequency Response Analysis Potentiostatic or
galvanostatic measurements 11
Slide 12
Advantages and Disadvantages Electrochemical Impedance
Spectroscopy (EIS) is also called AC Impedance or just Impedance
Spectroscopy. The usefulness of impedance spectroscopy lies in the
ability to distinguish the dielectric and electric properties of
individual contributions of components under investigation. For
example, if the behavior of a coating on a metal when in salt water
is required, by the appropriate use of impedance spectroscopy, a
value of resistance and capacitance for the coating can be
determined through modeling of the electrochemical data. The
modeling procedure uses electrical circuits built from components
such as resistors and capacitors to represent the electrochemical
behavior of the coating and the metal substrate. Changes in the
values for the individual components indicate their behavior and
performance. Impedance spectroscopy is a non-destructive technique
and so can provide time dependent information about the properties
but also about ongoing processes such as corrosion or the discharge
of batteries and e.g. the electrochemical reactions in fuel cells,
batteries or any other electrochemical process. 12
Slide 13
Advantages and Disadvantages Advantages. 1. Useful on high
resistance materials such as paints and coatings. 2. Time dependent
data is available 3. Non- destructive. 4. Quantitative data
available. 5. Use service environments. 6. System in thermodynamic
equilibrium 7. Measurement is small perturbation (approximately
linear) 8. Different processes have different time constants 9.
Large frequency range, Hz to GHz (and up) 10. Generally analytical
models available 11.Pre-analysis (subtraction procedure) leads to
plausible models and starting values Disadvantages. 1.Rather
expensive equipment, 2.Low frequencies difficult to measure 3.
Complex data analysis for quantification. 13
Slide 14
AC Circuit Theory and Representation of Complex Impedance
Values Concept of complex impedance: from R to Z Ohm's law defines
resistance in terms of the ratio between voltage E and current I :
The relationship is limited to only one circuit element -- the
ideal resistor. An ideal resistor has several simplifying
properties: It follows Ohm's Law at all current and voltage levels
It's resistance value is independent of frequency. AC current and
voltage signals though a resistor are in phase with each other
14
Slide 15
AC Circuit Theory and Representation of Complex Impedance
Values In practice circuit elements exhibit much more complex
behavior. This forces one to abandon the simple concept of a
resistance only. In its place we use impedance, which is a more
general circuit parameter (Z instead of R). Like resistance R,
impedance Z is a measure of the ability of a circuit to resist the
flow of electrical current. Electrochemical impedance is usually
measured by applying an AC potential (current) to an
electrochemical cell and measuring the current (voltage) through
the cell. Suppose that we apply a sinusoidal potential excitation.
The response to this potential is an AC current signal, containing
the excitation frequency and it's harmonics. This current signal
can be analyzed as a sum of sinusoidal functions (for a linear
system-see earlier). The current response to a sinusoidal potential
will be a sinusoid at the same frequency but shifted in phase.
15
Slide 16
AC Circuit Theory and Representation of Complex Impedance
Values The excitation signal, expressed as a function of time, has
the form of: E(t) is the potential at time t, E o is the amplitude
of the signal, and is the radial frequency. The relationship
between radial frequency (expressed in radians/second) and
frequency f (expressed in Hertz (1/sec). 16
Slide 17
AC Circuit Theory and Representation of Complex Impedance
Values In a linear system, the response signal, the current I(t),
is shifted in phase ( ) and has a different amplitude, I 0 : An
expression analogous to Ohm's Law allows us to calculate the
impedance (=the AC resistance) of the system : The impedance is
therefore expressed in terms of a magnitude, Z 0, and a phase shift
. This impedance may also be written as a complex function (see
next slide) : 17
Slide 18
AC Circuit Theory and Representation of Complex Impedance
Values 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: E(t) = E 0 exp(i t) 18
Slide 19
AC Circuit Theory and Representation of Complex Impedance
Values E(t) = E 0 cos( t), =2 f I(t) = I 0 cos( t- ) Or, if one
writes in complex notation: E(t) = E 0 exp(i t) I(t) = I 0 exp(i t
- i ) Z(t) = Z 0 exp(i ) = Z 0 (cos + isin ) E/I response for a
resistor ( =0) E/I response for a capacitor ( =-90) E/I response
for an inductor ( =90) 19
Slide 20
Data Presentation: Nyquist Plot with Impedance Vector The
expression for Z( ) is composed of a real and an imaginary part. If
the real part is plotted on the X axis and the imaginary part on
the Y axis of a chart, we get a "Nyquist plot (also Cole-Cole
plot). Notice that in this plot the y-axis is negative and that
each point on the Nyquist plot is the impedance Z at one frequency.
In the Nyquist plot the impedance is represented as a vector of
length |Z|. The angle between this vector and the x-axis is the
phase angle . Nyquist plots have one major shortcoming. When you
look at any data point on the plot, you cannot tell what frequency
was used to record that point. Low frequency data are on the right
side of the plot and higher frequencies are on the left. Y =1/Z. A
semicircle is characteristic of a single "time constant".
Electrochemical Impedance plots often contain several time
constants. Often only a portion of one or more of their semicircles
is seen. 20
Slide 21
The magnitude of Z and phase angle are given by the following,
respectively (with R = real and X c = imaginary, also a and b later
in the class): The impedance Z is a kind of a generalized
resistance R. The phase angle expresses the balance between
capacitive and resistance components in the series circuit. For a
pure resistance, =0; for a pure capacitance, = /2; and for
mixtures, intermediate phase angles are observed. AC Circuit Theory
and Representation of Complex Impedance Values 21
Slide 22
For a pure resistance R, E=IR, and the phase is zero. For a
pure capacitance C: Where X c is the capacitive reactance, 1/ C A
comparison of R and X c shows that X c must carry the dimensions of
a resistance, but the magnitude of X c falls with increasing
frequency. Data Presentation: Nyquist Plot with Impedance Vector
22
Slide 23
Take a look at the properties of a capacitor: Charge stored
(Coulombs): Change of voltage results in current, I: Alternating
voltage (ac): Impedance: Admittance: Data Presentation: Nyquist
Plot with Impedance Vector 23
Slide 24
Impedance resistance Admittance conductance: Data Presentation:
Nyquist Plot with Impedance Vector Representation of impedance
value, Z = a +jb, in the complex plane (see also
http://math.tutorvista.com/number-system/absolute-value-of-a-complex-
number.html 24
Slide 25
Data Presentation: Nyquist Plot with Impedance Vector What is
the impedance of an -R-C- circuit? Admittance? Semi- circle time
constant: = RC time constant: = RC 25
Slide 26
The parallel combination of a resistance and a capacitance,
start in the admittance representation: Transform to impedance
representation: A semicircle in the impedance plane! Data
Presentation: Nyquist Plot with Impedance Vector R C 26
Slide 27
Data Presentation: Nyquist Plot with Impedance Vector 1 MHz1 Hz
518 Hz R = 100 k C = 3 nF f max = 1/(6.3x3 10 -9 x10 5 )=530 Hz
27
Slide 28
Data Presentation: Nyquist Plot with Impedance Vector What
happens for > ? > : This is best observed in a so-called Bode
plot log(Z re ), log(Z im ) vs. log(f ), or log|Z| and phase vs.
log(f ) 28
Slide 29
Bode plot (Z re, Z im ) -1 -2 29
Slide 30
Bode, abs(Z), phase 30
Slide 31
The Bode Plot Another popular presentation method is the "Bode
plot". The impedance is plotted with log frequency on the x-axis
and both the absolute value of the impedance (|Z| =Z 0 ) and
phase-shift on the y-axis. The Bode plot for the RC circuit is
shown below. Unlike the Nyquist plot, the Bode plot explicitly
shows frequency information. 31
Slide 32
Data Presentation: Nyquist Plot with Impedance Vector 32 An
electrical layer of a device can often be described by a resistor
and capacitor in parallel: Voigt network.
Slide 33
When we plot the real and imaginary components of impedance in
the complex plane (Argand diagram), we obtain a semicircle or
partial semicircle for each parallel RC Voigt network: Nyquist plot
or also Cole-Cole plot. The diameter corresponds to the resistance
R. The frequency at the 90 position corresponds to 1/ = 1/RC 33
Data Presentation: Nyquist Plot with Impedance Vector
Slide 34
34 The Randles cell is one of the simplest and most common cell
models. It includes a solution resistance, a double layer capacitor
and a charge transfer or polarization resistance. In addition to
being a useful model in its own right, the Randles cell model is
often the starting point for other more complex models. The
equivalent circuit for the Randles cell is shown in the Figure. The
double layer capacity is in parallel with the impedance due to the
charge transfer reaction The Nyquist plot for a Randles cell is
always a semicircle. The solution resistance can found by reading
the real axis value at the high frequency intercept. This is the
intercept near the origin of the plot. This plot was generated
assuming that R s = 20 and Rp= 250 . The real axis value at the
other (low frequency) intercept is the sum of the polarization
resistance and the solution resistance. The diameter of the
semicircle is therefore equal to the polarization resistance (in
this case 250 ).
Slide 35
Data Presentation: Nyquist Plot with Impedance Vector 35
Slide 36
Data Presentation: Nyquist Plot with Impedance Vector 36
Slide 37
Data Presentation: Nyquist Plot with Impedance Vector 37
Slide 38
Summary Nyquist and Bode Plots 38
Slide 39
Review of Circuit Elements 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 39
Slide 40
Review of Circuit Elements EIS data is commonly analyzed by
fitting it to an equivalent electrical circuit model. Most of the
circuit elements in the model are common electrical elements such
as resistors, capacitors, and inductors. To be useful, the elements
in the model should have a basis in the physical electrochemistry
of the system. As an example, most models contain a resistor that
models the cell's solution resistance. Some knowledge of the
impedance of the standard circuit components is therefore very
important. The Table below lists the common circuit elements, the
equation for their current versus voltage relationship, and their
impedance: ComponentCurrent Vs.VoltageImpedance resistorE= IRZ = R
inductorE = L di/dt Z = i L capacitorI = C dE/dt Z = 1/i C Notice
that the impedance of a resistor is independent of frequency and
has only a real component. Because there is no imaginary impedance,
the current through a resistor is always in phase with the voltage.
The impedance of an inductor increases as frequency increases.
Inductors have only an imaginary impedance component. As a result,
an inductor's current is phase shifted 90 degrees with respect to
the voltage. The impedance versus frequency behavior of a capacitor
is opposite to that of an inductor. A capacitor's impedance
decreases as the frequency is raised. Capacitors also have only an
imaginary impedance component. The current through a capacitor is
phase shifted - 90 degrees with respect to the voltage. 40
Slide 41
Review of Circuit Elements Resistance and impedance both go up
when resistors are combined in series. Now suppose that we connect
two 2 F capacitors in series. The total capacitance of the combined
capacitors is 1 F Suppose we have a 1 and a 4 resistor is series.
The impedance of a resistor is the same as its resistance. We thus
calculate the total impedance Z eq : Impedance goes up, but
capacitance goes down when capacitors are connected in series. This
is a consequence of the inverse relationship between capacitance
and impedance. C1C1 C2C2 R1R1 R2R2 41
Slide 42
Resistance: ZR E I 0 Capacitance: Z C 1 I E 90 Review of
Circuits Elements 42
Slide 43
In a general sense, we ought to be able to represent the
performance of a cell by an equivalent circuit of resistors and
capacitors under a given excitation. The elements of equivalent
circuit of a cell: double-layer capacitance C d, faradaic impedance
Z f, solution resistance R s, charge transfer resistance R ct,
Warburg impedance Z w. Equivalent Circuit of a Cell 43
Slide 44
Electrolyte resistance R is often a significant factor in the
impedance of an electrochemical cell. A modern 3 electrode
potentiostat compensates for the solution resistance between the
counter and reference electrodes. However, any solution resistance
between the reference electrode and the working electrode must be
considered when you model your cell. The resistance of an ionic
solution depends on the ionic concentration, type of ions,
temperature and the geometry of the area in which current is
carried. In a bounded area with area A and length l carrying a
uniform current the resistance is defined as: 44 Equivalent Circuit
of a Cell: R s and C d
Slide 45
Standard chemical handbooks list values for specific solutions.
For other solutions and solid materials, you can calculate from
specific ion conductances. The units for are Siemens per meter
(S/m). The Siemens is the reciprocal of the ohm, so 1 S = 1/ohm
Unfortunately, most electrochemical cells do not have uniform
current distribution through a definite electrolyte area. The major
problem in calculating solution resistance therefore concerns
determination of the current flow path and the geometry of the
electrolyte that carries the current. A comprehensive discussion of
the approaches used to calculate practical resistances from ionic
conductances is beyond the scope of this class. Fortunately, you
don't usually calculate solution resistance from ionic
conductances. Instead, it is found when you fit a model to
experimental EIS data. 45
Slide 46
A Resistance and capacitance in series f is low: f is high: In
electrochemical cell: R=R s : solution resistance C=C d : double
layer capacitance Electrochemical Impedance Spectroscopy Equivalent
Circuit of a Cell: R s and C d 46
Slide 47
A electrical double layer exists at the interface between an
electrode and its surrounding electrolyte. This double layer is
formed as ions from the solution "stick on" the electrode surface.
Charges in the electrode are separated from the charges of these
ions. The separation is very small, on the order of angstroms.
Charges separated by an insulator form a capacitor. On a bare metal
immersed in an electrolyte, you can estimate that there will be
approximately 30 F of capacitance for every cm 2 of electrode area.
The value of the double layer capacitance depends on many variables
including electrode potential, temperature, ionic concentrations,
types of ions, oxide layers, electrode roughness, impurity
adsorption, etc. Principle of the Electric Double-Layer: Here C
electrodes 47 Equivalent Circuit of a Cell: R s and C d
Slide 48
Equivalent Circuit of a Cell: CPE Constant Phase Element (for
double layer capacity in real electrochemical cells) Capacitors in
EIS experiments often do not behave ideally. Instead, they act like
a constant phase element (CPE) as defined below. When this equation
describes a capacitor, the constant A = 1/C (the inverse of the
capacitance) and the exponent = 1. For a constant phase element,
the exponent a is less than one. The "double layer capacitor" on
real cells often behaves like a CPE instead of like a capacitor.
Several theories have been proposed to account for the non-ideal
behavior of the double layer but none has been universally accepted
(fractal explanation!). In most cases, you can safely treat as an
empirical constant and not worry about its physical basis. 48
Slide 49
Equivalent Circuit of a Cell: CPE Constant Phase Element: Y CPE
= Y 0 n {cos(n /2) + j sin(n /2)} n = 1 Capacitance:C = Y 0 n =
Warburg: = Y 0 n = 0 Resistance:R = 1/Y 0 n = -1 Inductance:L = 1/Y
0 All other values, fractal? Non-ideal capacitance, n < 1
(between 0.8 and 1?) 49
Slide 50
Equivalent Circuit of a Cell: CPE Deviation from ideal
dispersion: Constant Phase Element (CPE), (symbol: Q ) General
observations: Semicircle (RC ) depressed vertical spur (C )
inclined Warburg less than 45 n = 1, , 0, -1, ? n = 1, , 0, -1, ?
50
Slide 51
Equivalent Circuit of a Cell: CPE How to explain this non-ideal
behaviour? 1980s: 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 51
Slide 52
Equivalent Circuit of a Cell: CPE Fractal line Sierpinski
carpet Self similarity! 52
Slide 53
Mixed kinetic and diffusion control C dl or CPE RPRP R ZWZW
with 0 n 1 Equivalent Circuit of a Cell 53
Slide 54
Equivalent Circuit of a Cell: R p(or ct) and C d 54 A charge
transfer resistance is formed by common kinetically controlled
electrochemical reaction Consider a metal substrate in contact with
an electrolyte. The metal molecules can electrolytically dissolve
into the electrolyte, according to: or more generally: 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: i o =
exchange current density C o = concentration of oxidant at the
electrode surface C o * = concentration of oxidant in the bulk C R
= concentration of reductant at the electrode surface F = Faradays
constant T = temperature R = gas constant a = reaction order n =
number of electrons involved = overpotential ( E - E 0 )
Slide 55
Equivalent Circuit of a Cell: R p(or ct) and C d The
overpotential, , 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, C o =C o * and C R =C R *. This simplifies the
last equation into: This equation is called the Butler-Volmer
equation. It is applicable when the polarization depends only on
the charge transfer kinetics. Stirring will minimize diffusion
effects and keep the assumptions of C o =C o * and C R =C R *
valid. When the overpotential, , 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 i 0 density can be calculated when R ct is known
(see R c t in next figure). 55
Slide 56
A resistance and capacitance in parallel (Randles circuit) Z=R
s at high frequency Z=R ct +R s at low frequency Equivalent Circuit
of a Cell: R p(or ct) and C d 56
Slide 57
Equivalent Circuit of a Cell: R p(or ct) and C d 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 Butler-Volmer equation). I is the measured
cell current in amps, I corr is the corrosion current in amps, E oc
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 (E-E oc is
small) to the buler Volmer equation, we get the following: which
introduces a new parameter, R p, the polarization resistance. If
the Tafel constants i are known, you can calculate the I corr from
R p. The I corr in turn can be used to calculate a corrosion rate.
The R p parameter comes again from the Nyquist plot. 57
Slide 58
Equivalent Circuit of a Cell: R p(or ct) and C d 58
Slide 59
Equivalent Circuit of a Cell: Warburg Impedance 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, is the Warburg
coefficient defined as: = radial frequency D O = diffusion
coefficient of the oxidant D R = diffusion coefficient of the
reductant A = surface area of the electrode n = number of electrons
transferred C* = bulk concentration of the diffusing species
(moles/cm 3 ) 59
Slide 60
Define impedance in Laplace space! Equivalent Circuit of a
Cell: Warburg Impedance Take the Laplace variable, p, complex: p =
s + j . Steady state: s 0, which yields the impedance: In solution:
60
Slide 61
Equivalent Circuit of a Cell: Warburg Impedance The former
equation 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: with, = Nernst diffusion layer thickness D = an average value
of the diffusion coefficients of the diffusing species This more
general equation is called the "finite" Warburg. For high
frequencies where , or for an infinite thickness of the diffusion
layer where d , this equation becomes the infinite Warburg
impedance. 61
Slide 62
Equivalent Circuit of a Cell: Warburg Impedance semicircle 45
62
Slide 63
Equivalent Circuit of a Cell: Warburg ImpedanceCoating
Capacitor A capacitor is formed when two conducting plates are
separated by a non-conducting 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: Material
rr vacuum1 water80.1 ( 20 C ) organic coating 4 - 8 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
63
Slide 64
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 cm
2 sample, a 25 m coating, and r = 6 ) f i = 0.1 Hz (lowest scan
frequency -- a bit higher than typical) f f = 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 10 10 .
This is close to the limit of measurement of most EIS systems
Equivalent Circuit of a Cell: Warburg ImpedanceCoating Capacitor
64
Slide 65
Equivalent Circuit of a Cell: Warburg ImpedanceCoating
Capacitor 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 65
Slide 66
Equivalent Circuit of a Cell: Warburg ImpedanceCoating
Capacitor Classification of types of capacitances sourceapproximate
value geometric2-20 pF(cm -1 ) grain boundaries1-10 nF(cm -1 )
double layer / space charge0.1-10 F/cm 2 surface charge /adsorbed
species0.2 mF/cm 2 (closed) pores1-100 F/cm 3 pseudo capacitances
stoichiometry changeslarge !!!! 66
Slide 67
Equivalent Circuit of a Cell: Warburg ImpedancePorous Coating
67
Slide 68
Summary 68
Slide 69
Summary Below we show some common equivalent circuits models.
Equations for both the admittance and impedance are given for each
element. Equivalent element AdmittanceImpedance R 1/RR C iCiC1/1/i
C L 1/i LiLiL W (infinite Warburg) Y 0 (i ) 1/2 1/Y 0 (i ) 1/2 O
(finite Warburg) Q (CPE) Y 0 (i ) 1/Y 0 (i ) 69
Slide 70
Summary By using the various Cole-Cole 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 70
Slide 71
Note : Data validation Kramers-Kronig relations Kramers-Kronig
conditions: causality linearity stability (finiteness) Real and
imaginary parts are linked through the K-K transforms: Response
only due to input signal Response scales linearly with input signal
State of system may not change during measurement 71
Slide 72
Note: Putting K-K in practice Problem: Finite frequency range:
extrapolation of dispersion assumption of a model. Relations, Real
imaginary: Imaginary real: not a singularity! 72