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Electrochemical Impedance Spectroscopy Theory: A Primer

• About The EIS Theory Primer

  • Definition of Impedance

  • Data Presentation

  • Electrochemistry - A Linear System?

  • Steady State Systems

  • Time and Frequency Domains and Transforms

  • Electrical Circuit Elements

  • Serial and Parallel Combinations of Circuit Elements

• Physical Electrochemistry and Equivalent Circuit Elements

  • Electrolyte Resistance

  • Double Layer Capacitance

  • Polarization Resistance

  • Charge Transfer Resistance

  • Diffusion

  • Coating Capacitance

  • Constant Phase Element

  • Virtual Inductor

• Common Equivalent Circuit Models

  • Model #1 -- A Purely Capacitive Coating

  • Model #2 -- Randles Cell

  • Model #3 -- Mixed Kinetic and Diffusion Control

  • Model #4 -- Coated Metal

• Extracting Model Parameters from Data

  • Modeling Overview

  • Non-linear Least Squares Fitting

  • Uniqueness of Models

  • Literature

• Kramers-Kronig  Analysis

About The EIS Theory Primer

The EIS Theory Primer presents an introduction to Electrochemical Impedance Spectroscopy (EIS) theory. This application note has been kept as free from mathematics and electrical theory as is possible. If you still find the material presented here difficult to understand, don't stop reading. You will get useful information from this application note even if you don't follow all of the discussions.

Five major topics are covered in this application note:

No prior knowledge of electrical circuit theory or electrochemistry is assumed. Each topic starts out at a quite elementary level. Each topic proceeds from this simple beginning to cover more advanced material.

Almost everyone knows about the concept of electrical resistance. It is the ability of a circuit element to resist the flow of electrical current. Ohm's law (Equation 2-1) defines resistance in terms of the ratio between voltage E and current I.

     (2-1)

While this is a well known relationship, it's use is limited to only one circuit element -- the ideal resistor. An ideal resistor has several simplifying properties:

The real world contains circuit elements that exhibit much more complex behavior. These elements force us to abandon the simple concept of resistance. In its place we use impedance, which is a more general circuit parameter. Like resistance, impedance is a measure of the ability of a circuit to resist the flow of electrical current. Unlike resistance, impedance is not limited by the simplifying properties listed above.

Electrochemical impedance is usually measured by applying an AC potential to an electrochemical cell and measuring the current 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 (a Fourier series).

Electrochemical Impedance is normally measured using a small excitation signal. This is done so that the cell's response is pseudo-linear. Linearity is described in more detail in a following section. In a linear (or pseudo-linear) system, the current response to a sinusoidal potential will be a sinusoid at the same frequency but shifted in phase. See Figure 2-1.

 

Figure 2-1

Sinusoidal Current Response in a Linear System

The excitation signal, expressed as a function of time, has the form

  (2-2)

E(t) is the potential at time t_r E_o is the amplitude of the signal, and w is the radial frequency. The relationship between radial frequency w (expressed in radians/second) and frequency f (expressed in hertz) is:

   (2-3)

In a linear system, the response signal, I_t, is shifted in phase () and has a different amplitude, I₀:

   (2-4)

An expression analogous to Ohm's Law allows us to calculate the impedance of the system as:

     (2-5)

The impedance is therefore expressed in terms of a magnitude, Z₀, and a phase shift, f.

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. See Figure 2-2. 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 and frequency response analyzers.

 

Using Eulers relationship,

   (2-6)

it is possible to express the impedance as a complex function. The potential is described as,

   (2-7)

and the current response as,

   (2-8)

The impedance is then represented as a complex number,

   (2-9)

 

Data Presentation

Look at Equation 2-9 in the previous section. The expression for Z(w) 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". See Figure 2-3. Notice that in this plot the y-axis is negative and that each point on the Nyquist plot is the impedance at one frequency.

 

Figure 2-3

Nyquist Plot with Impedance Vector

Figure 2-3 has been annotated to show that low frequency data are on the right side of the plot and higher frequencies are on the left. This is true for EIS data where impedance usually falls as frequency rises (this is not true of all circuits).

On the Nyquist plot the impedance can be represented as a vector of length |Z|. The angle between this vector and the x-axis is f.

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.

The Nyquist plot in Figure 2-3 results from the electrical circuit of Figure 2-4. The 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.

 

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| =Z₀ ) and phase-shift on the y-axis.

The Bode plot for the electric circuit of Figure 2-4 is shown in Figure 2-5. Unlike the Nyquist plot, the Bode plot explicitly shows frequency information.

 

Electrochemistry - A Linear System?

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.

The following definition of a linear system is taken from Signals and Systems by Oppenheim and Willsky:

For a potentiostated electrochemical cell, the input is the potential and the output is the current. Electrochemical cells are not linear!  Doubling the voltage will not necessarily double the current.

However, Figure 2-6 shows how 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.

 

If the system is non-linear, the current response will contain harmonics of the excitation frequency.

Some researchers have made use of this phenomenon. Linear systems should not generate harmonics, so the presence or absence of significant harmonic response allows one to determine the system's linearity. Other researchers have intentionally used larger amplitude excitation potentials. They use the harmonic response to estimate the curvature in the cell's current voltage curve.

 

Steady State Systems

Measuring an EIS spectrum takes time (often many hours). The system being measured must be at a steady state throughout the time required to measure the EIS spectrum. A common cause of problems in EIS measurements and their analysis is drift in the system being measured.

In practice a steady state can be difficult to achieve. The cell can change through adsorption of solution impurities, growth of an oxide layer, build up of reaction products in solution, coating degradation, temperature changes, to list just a few factors.

Standard EIS analysis tools may give you wildly inaccurate results on a system that is not at a steady state.

 

Time and Frequency Domains and Transforms

Signal processing theory refers to data domains. The same data can be represented in different domains. In EIS, we use two of these domains, the time domain and the frequency domain.

In the time domain, signals are represented as signal amplitude versus time. Figure 2-7 demonstrates this for a signal consisting of two superimposed sine waves.

 

Figure 2-8 shows the same data in the frequency domain. The data is plotted as amplitude versus frequency.

 

You use the Fourier transform and inverse Fourier transform to switch between the domains. The common term, FFT, refers to a fast, computerized implementation of the Fourier transform. Detailed discussion of these transforms is beyond the scope of this manual. Several of the references given at the end of this chapter contain more information on the Fourier transform and its use in EIS.

In modern EIS systems, lower frequency data are usually measured in the time domain. The controlling computer applies a digital approximation to a sine wave to the cell by means of a digital to analog converter. The current response is measured using an analog to digital computer. An FFT is used to convert the current signal into the frequency domain.

 

Electrical 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 quite useful. Table 2-1 lists the common circuit elements, the equation for their current versus voltage relationship, and their impedance.

 

Common Electrical Elements

 

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.

 

Serial and Parallel Combinations 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.

Fortunately, there are simple formulas that describe the impedance of circuit elements in both parallel and series combinations.

 

For linear impedance elements in series you calculate the equivalent impedance from:

    (2-10)

 

For linear impedance elements in parallel you calculate the equivalent impedance from:

(2-11)

We will calculate two examples to illustrate a point about combining circuit elements. Suppose we have a 1 and a 4 resistor is series. The impedance of a resistor is the same as its resistance (see Table 2-1). We thus calculate the total impedance as:

Resistance and impedance both go up when resistors are combined in series.

Now suppose that we connect two 2 µF capacitors in series. The total capacitance of the combined capacitors is 1 µF.

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.

Solution resistance 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:

   (2-12)

where r is the solution resistivity. The conductivity of the solution, k , is more commonly used in solution resistance calculations. Its relationship with solution resistance is:

    (2-13)

Standard chemical handbooks list k values for specific solutions. For other solutions, you can calculate k from specific ion conductances. The units for k 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 well beyond the scope of this manual.

Fortunately, you don't usually calculate solution resistance from ionic conductances. Instead, it is found when you fit a model to experimental EIS data.

 

Double Layer Capacitance

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 µF of capacitance for every cm² 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.

 

Polarization Resistance

Whenever the potential of an electrode is forced away from it's value at open circuit, that is referred to as polarizing the electrode. When an electrode is polarized, it can cause current to flow via electrochemical reactions that occur at the electrode surface. The amount of current is controlled by the kinetics of the reactions and the diffusion of reactants both towards and away from the electrode.

In cells where an electrode undergoes uniform corrosion at open circuit, the open circuit potential is controlled by the equilibrium between two different electrochemical reactions. One of the reactions generates cathodic current and the other anodic current. The open circuit potential ends up at the potential where the cathodic and anodic currents are equal. It is referred to as a mixed potential. The value of the current for either of the reactions is known as the corrosion current.

Mixed potential control also occurs in cells where the electrode is not corroding. While this section discusses corrosion reactions, modification of the terminology makes it applicable in non-corrosion cases as well.

When there are two simple, kinetically controlled reactions occurring, the potential of the cell is related to the current by the following (known as the Butler-Volmer equation).

 

         (2-14)

where

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,
b_a is the anodic Beta coefficient in volts/decade,
b_c is the cathodic Beta coefficient in volts/decade.

 

If we apply a small signal approximation (E-Eoc is small) to equation 2-14, we get the following:

                    (2-15)

which introduces a new parameter, R_p, the polarization resistance. As you might guess from its name, the polarization resistance behaves like a resistor.

If the Tafel constants are known, you can calculate the I_corr from R_p using equation 2-15. I_corr in turn can be used to calculate a corrosion rate.

We will further discuss the R_p parameter when we discuss cell models.

 

Charge Transfer Resistance

A similar resistance is formed by a single kinetically controlled electrochemical reaction. In this case we do not have a mixed potential, but rather a single reaction at equilibrium.

Consider a metal substrate in contact with an electrolyte. The metal molecules can electrolytically dissolve into the electrolyte, according to:

(2-16)

or more generally:

(2-17)

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 is:

      (2-18)

with,

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
 
h = 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, C_o=C_o* and C_R=C_R*. This simplifies equation 2-18 into:

     (2-19)

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, h, is very small and the electrochemical system is at equilibrium, the expression for the charge transfer resistance changes into:

      (2-20)

From this equation the exchange current density can be calculated when R_ct is known.

 

Diffusion

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 is:

      (2-21)

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 equation 2-21, s is the Warburg coefficient defined as:

        (2-22)

In which,

w = 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/cm³)

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:

          (2-23)

with,

d = 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 w ® ¥ , or for an infinite thickness of the diffusion layer where d ® ¥ , equation 2-23 simplifies to the infinite Warburg impedance.

 

Coating Capacitance

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:

          (2-24)

With,

e_o = electrical permittivity

e_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. Table 2-2 gives you a few useful e_r values.

 

Table 2-2

Typical Relative Electrical Permittivities

 

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.

 

Constant Phase Element

Capacitors in EIS experiments often do not behave ideally. Instead, they act like a constant phase element (CPE) as defined below.

The impedance of a capacitor has the form:

         (2-25)

When this equation describes a capacitor, the constant A = 1/C (the inverse of the capacitance) and the exponent a = 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 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. In most cases, you can safely treat a as an empirical constant and not worry about its physical basis.

 

Virtual Inductor

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, cell lead inductance and potentiostat non-idealities. In these cases, it represents an error in the EIS measurement.

 

Common Equivalent Circuit Models

In the following section we show some common equivalent circuits models. These models can be used to interpret simple EIS data. Many of these models have been included as standard models in the Echem Analyst^TM EIS300 analysis package.

To elements used in the following equivalent circuits are presented in Table 2-3. Equations for both the admittance and impedance are given for each element.

 

Model #1 -- A Purely Capacitive Coating

A metal covered with an undamaged coating generally has a very high impedance. The equivalent circuit for such a situation is in Figure 2-11.

 

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 figure 2-12. 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 e_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 10¹⁰ . This is close to the limit of measurement of most EIS systems.

The same data are shown in a Bode plot in Figure 2-13. 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.

 

Figure 2-13

Typical Bode Plot for an Excellent Coating

Water uptake into the film is usually a fairly slow process. It can be measured by taking EIS spectra at set time intervals. An increase in the film capacitance can be attributed to water uptake.

 

Model #2 -- Randles Cell

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 Figure 2-14. The double layer capacity is in parallel with the impedance due to the charge transfer reaction.

 

Figure 2-14

Randles Cell Schematic Diagram

Figure 2-15 is the Nyquist plot for a typical Randles cell. The parameters in this plot were calculated assuming a 1 cm² electrode undergoing uniform corrosion at a rate of 1 mm/year. Reasonable assumptions were made for the b coefficients, metal density and equivalent weight. The polarization resistance under these conditions calculated out to 250 . A capacitance of 40 µF/cm² and a solution resistance of 20 were also assumed.

 

Figure 2-15

Nyquist Plot for 1 mm/year Corrosion Rate

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. Remember 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 ).

Figure 2-16 is the Bode plot for the same cell. The solution resistance and the sum of the solution resistance and the polarization resistance can be read from the magnitude plot. The phase angle does not reach 90^° as it would for a pure capacitive impedance. If the values for R_s and Rp were more widely separated the phase would approach 90^°.

 

Figure 2-16

Bode Plot for 1 mm/year Corrosion Rate

Model #3 -- Mixed Kinetic and Diffusion Control

First consider a cell where semi-infinite diffusion is the rate determining step, with a series solution resistance as the only other cell impedance.

A Nyquist plot for this cell is shown in Figure 2-17. R_s was assumed to be 20 W. The Warburg coefficient calculated to be about 120 sec^-1/2 at room temperature for a two electron transfer, diffusion of a single species with a bulk concentration of 100 µM and a typical diffusion coefficient of 1.6 x10^-5 cm²/sec.     Notice that the Warburg Impedance appears as a straight line with a slope of 45°.

 

Figure 2-17

Nyquist Plot for a Warburg Impedance

The same data is plotted in the Bode format in Figure 2-18. The phase angle of a Warburg impedance is 45°.

 

Figure 2-18

Bode Plot for a Warburg Impedance

Adding a double layer capacitance and a charge transfer impedance, we get the equivalent circuit in Figure 2-19.

 

Figure 2-19

Equivalent Circuit with Mixed Kinetic and Charge Transfer Control

This circuit models a cell where polarization is due to a combination of kinetic and diffusion processes. The Nyquist plot for this circuit is shown in Figure 2-20. As in the above example, the Warbug coefficient is assumed to be  about 150 W sec^-1/2. Other assumptions: R_s = 20 , R_ct = 250 , and C_dl = 40 µF.

 

The Bode plot for the same data is shown in Figure 2-21. The lower frequency limit was moved down to 1mHz to better illustrate the differences in the slope of the magnitude and in the phase between the capacitor and the Warburg impedance. Note that the phase approaches 45° at low frequency.

 

Model #4 -- Coated Metal

The impedance behavior of a purely capacitive coating was discussed above. Most coatings degrade with time, resulting in more complex behavior.

After a certain amount of time, water penetrates into the coating and forms a new liquid/metal interface under the coating. Corrosion phenomena can occur at this new interface.

The impedance of coated metals has been very heavily studied. The interpretation of impedance data from failed coatings can be very complicated. Only the simple equivalent circuit shown in Figure 2-22 will be discussed here.

Even this simple model has been the cause of some controversy in the literature. Most researchers agree that this model can be used to evaluate the quality of a coating. However, they do not agree on the physical processes that create the equivalent circuit elements. The discussion below is therefore only one of several interpretations of this model.

The capacitance of the intact coating is represented by C_c. Its value is much smaller than a typical double layer capacitance. Its units are pF or nF, not µF. R_po (pore resistance) is the resistance of ion conducting paths that develop in the coating. These paths may not be physical pores filled with electrolyte.

On the metal side of the pore, we assume that an area of the coating has delaminated and a pocket filled with an electrolyte solution has formed. This electrolyte solution can be very different from the bulk solution outside of the coating. The interface between this pocket of solution and the bare metal is modeled as a double layer capacity in parallel with a kinetically controlled charge transfer reaction.

 

Figure 2-22

Equivalent Circuit for a Failed Coating

When you use EIS to test a coating, you fit a data curve to this type of model. The fit estimates values for the model's parameters, such as the pore resistance or the double layer capacitance. You then use these parameters to evaluate the degree to which the coating has failed.

In order to show a realistic data curve, we need to do this operation in reverse. Assume that we have a 10 cm² sample of metal coated with a 12 mm film and that we have 5 delaminated areas. 1% of the total metal area is delaminated. The pores in the film that access these delaminated areas are represented as solution filled cylinders with a 30 mm diameter.

The parameters used to develop the curves are shown below:

C_c = 4 nF Calculated for 10 cm² area , e_r = 6 and 12 µm thickness
R_po = 3400 Calculated assuming k = 0.01 S/cm
R_s = 20 Assumed
C_dl = 4 mF Calculated for 1% of 10 cm² area and assuming 40 µF/cm²
R_ct = 2500 Calculated for 1% of 10 cm² area and using Polarization
Resistance assumptions from an earlier discussion

With these parameters, the Nyquist plot for this model is shown in Figure 2-23. Notice that there are two well defined time constants in this plot.

 

The Bode plot of the same data is shown in Figure 2-24. The two time constants are visible but less pronounced on this plot.

The Bode plot does not go high enough in frequency to measure the solution resistance. In practice this is not a problem, because the solution resistance is a property of the test solution and the test cell geometry, not a property of the coating. Therefore, it is usually not very interesting when you are testing coatings.

 

EIS data is generally analyzed in terms of an equivalent circuit model. The analyst tries to find a model whose impedance matches the measured data.

The type of electrical components in the model and their interconnections controls the shape of the model's impedance spectrum. The model's parameters (i.e. the resistance value of a resistor) controls the size of each feature in the spectrum. Both these factors effect the degree to which the model's impedance spectrum matches a measured EIS spectrum.

In a physical model, each of the model's components is postulated to come from a physical process in the electrochemical cell. All of the models discussed earlier in this chapter are physical models. The choice of which physical model applies to a given cell is made from knowledge of the cell's physical characteristics. Experienced EIS analysts use the shape of a cell's EIS spectrum to help choose among possible physical models for that cell.  For an excellent discussion on fitting a physical model to your EIS data, see the Application Note on Equivalent Circuit Modeling.

Models can also be partially or completely empirical. The circuit components in this type of model are not assigned to physical processes in the cell. The model is chosen to given the best possible match between the model's impedance and the measured impedance.

An empirical model can be constructed by successively subtracting component impedances from a spectrum. If the subtraction of an impedance simplifies the spectrum, the component is added to the model, and the next component impedance is subtracted from the simplified spectrum. This process ends when the spectrum is completely gone (Z=0).

As we shall see, physical models are generally preferable to empirical models.

 

Non-linear Least Squares Fitting

Modern EIS analysis uses a computer to find the model parameters that cause the best agreement between a model's impedance spectrum and a measured spectrum. For most EIS data analysis software, a non-linear least squares fitting (NLLS) Levenberg-Marquardt algorithm is used.

NLLS starts with initial estimates for all the model's parameters which must be provided by the user. Starting from this initial point, the algorithm makes changes in several or all of the parameter values and evaluates the resulting fit. If the change improves the fit, the new parameter value is accepted. If the change worsens the fit, the old parameter value is retained. Next a different parameter value is changed and the test is repeated. Each trial with new values is called an iteration. Iterations continue until the goodness of fit exceeds an acceptance criterion, or until the number of iterations reaches a limit.

NLLS algorithms are not perfect. In some cases they do not converge on a useful fit. This can be the result of several factors including:

· An incorrect model for the data set being fitted.

· Poor estimates for the initial values.

· Noise

In addition, the fit from an NLLS algorithm can look poor when the fit's spectrum is superimposed on the data spectrum. It appears as though the fit ignores a region in the data. To a certain extent this is what happens. The NLLS algorithm optimizes the fit over the entire spectrum. It does not care if the fit looks poor over a small section of the spectrum.

In Gamry EIS300 Electrochemical Impedance Spectroscopy Software Version 4, a Simplex algorithm is provided in addition to the Levenberg-Marquardt.  The Simplex algorithm can sometimes converge with data that is troubling to the NLLS algorithm.

 

Uniqueness of Models

The impedance spectrum shown in Figure 2-25 shows two clearly defined time constants.

 

This spectrum can be modeled by any of the equivalent circuits shown in Figure 2-26.

 

As you can see, there is not a unique equivalent circuit that describes the spectrum. You cannot assume that an equivalent circuit that produces a good fit to a data set represents an accurate physical model of the cell.

Even physical models are suspect in this regard. Whenever possible, the physical model should be verified before it is used. One way to verify the model is to alter a single cell component (for example increase a paint layer thickness) and see if you get the expected changes in the impedance spectrum.

Empirical models should be treated with even more caution. You can always get a good looking fit by adding lots of circuit elements to a model. Unfortunately, these elements will have little relevance to the cell processes that you are trying to study. Drawing conclusions based on changes in these elements is especially dangerous. Empirical models should therefore use the fewest elements possible.

Kramers-Kronig Analysis

The Kramers-Kronig (K-K) relations can be used to evaluate data quality.  The K-K relations demand that causal, complex plane spectral data shows dependence between magnitude and phase.   The real part of a spectrum can be obtained by an integration of the imaginary part and vice versa.  

The K-K relations will always be true for EIS data that is linear, causal, and stable.  If measured real and imaginary spectral data to not comply with the K-K relations, the data must violate one of these conditions. 

Unfortunately, the K-K transform requires integration over a range of frequency from zero to infinity.  Since no one can measure spectral data over that range, evaluating the k-K relations via integration always involves assumptions about the behavior of a spectrum outside the frequency over which it was measured. 

In practice, K-K analysis is performed by fitting a generalized model to spectral data.  Agarwal et al¹ proposed use of a model consisting of m series connected Voigt elements:  -R-(RC)_m-.  A Voigt element is a resistor and capacitor connected in parallel.  The parameter m is generally equal to the number of complex plane data points in the spectrum.  This model is by definition K-K compliant.  If you can obtain a good fit of this model to measured data, the data must also be K-K compliant.  Boukamp², proposed a means for doing the fit via linear equations, eliminating possible non-convergence issues.   This is the approach taken in Gamry’s K-K fit within the Echem Analyst.

In the Gamry Echem Analyst, when you select Kramers-Kronig on an impedance menu a model of the type described above is fit to the selected region of the spectrum.   If the fit is poor, you can assume that the data is not K-K transformable and is therefore of poor quality.   There is little point fitting non-K-K compliant data to an equivalent circuit model. 

A tab in the Echem Analyst allows you to look at the goodness-of-fit and a plot of the residuals (difference between the fit and the data) versus frequency.   A pre-fit parameter allows you to select the number of Voigt elements in each decade of frequency.  Selection of a value smaller than the data density in the measured spectrum may improve the fit if the spectrum is noisy. 

(1)       P. Agarwal, M.E. Orazem and L.H. Garcia-Rubio, J. Electrochem. Soc, 139, 1917 (1992)

(2)          B.A. Boukamp, J. Electrochem. Soc, 142, 1885 (1995)

 

Literature

The following sources were used in preparing this chapter. The reader is encouraged to consult them for additional information.

Electrochemical Impedance and Noise,  Robert Cottis and Stephen Turgoose, NACE International, 1440 South Creek Drive, Houston, TX 77084-4906.  Phone: 281-228-6200.  Fax: 281-228-6300.  ISBN: 1-57590-093-9.

Impedance Spectroscopy; Theory, Experiment, and Applications, 2nd ed. , E. Barsoukov, J.R. Macdonald, eds., Wiley Interscience Publications, 2005.

Electrochemical Methods; Fundamentals and Applications, A.J. Bard, L.R. Faulkner, Wiley Interscience Publications 2000.

Electrochemical Impedance: Analysis and Interpretation, J.R. Scully, D.C. Silverman, and M.W. Kendig, editors, ASTM, 1993.

Physical Chemistry, P.W. Atkins, Oxford University Press ,1990.

Signals and Systems, A.V. Oppenheim and A.S. Willsky, Prentice-Hall, 1983.

The Use of Impedance Measurements in Corrosion Research; The Corrosion Behavior of Chromium and Iron Chromium Alloys, J.A.L. Dobbelaar, PhD thesis TU-Delft, 1990.

Characterization of Organic Coatings with Impedance Measurements; A Study of Coating Structure, Adhesion and Underfilm Corrosion, F. Geenen, PhD thesis, TU-Delft, 1990.

Identification of Electrochemical Processes by Frequency Response Analysis, C. Gabrielle, Solartron Instrumentation Group, 1980.

Comprehensive Treatise of Electrochemistry; Volume 9 Electrodics: Experimental Techniques; E. Yeager, J.O'M. Bockris, B.E. Conway, S. Sarangapani, Chapter 4 "AC Techniques", M. Sluyters-Rehbach, J.H. Sluyters, Plenum Press, 1984.

Mansfeld, F., "Electrochemical Impedance Spectroscopy (EIS) as a New Tool for Investigation Methods of Corrosion Protection", Electrochimica Acta, 35 (1990), 1533.

Walter, G.W., "A Review of Impedance Plot Methods Used for Corrosion Performance Analysis of Painted Metals", Corrosion Science, 26 (1986) 681.

Kendig, M., J. Scully, "Basic Aspects of Electrochemical Impedance Application for the Life Prediction of Organic Coatings on Metals", Corrosion, 46 (1990) 22.

Fletcher, S., “Tables of Degenerate Electrical Networks for Use in the Equivalent-Circuit Analysis of Electrochemical Systems”, J. Electrochem. Soc., 141 (1994) 1823.

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