Experiments > I - Electrochemical Frequency Modulation

Description

Electrochemical Frequency Modulation (EFM) is a nondestructive corrosion-measurement technique that can directly give values of the corrosion current without a priori knowledge of Tafel constants. Both a single-run EFM measurement and a long-term EFM Trend experiment are available in single-potentiostat or multiplexed-potentiostat variations. Like Electrochemical Impedance Spectroscopy (EIS), EFM is a technique using small AC signals. Unlike EIS, however, two sine waves (at different frequencies) are applied to the cell simultaneously. Because current is a non-linear function of potential (see Butler-Volmer), the system responds in a non-linear way to the potential excitation. The current response contains not only the input frequencies, but also contains frequency components which are the sum, difference, and multiples of the two input frequencies.

The theory is beyond the scope of this introduction. However, this non-linear response has been shown to contain enough information about the corroding system so that the corrosion current can be calculated directly.

The two frequencies may not be chosen at random. They must both be small, integer-multiples of a "base frequency" that determines the length of the experiment. The figure below shows the waveform when the two input frequencies are 2 Hz and 5 Hz. The base frequency was 1 Hz, so the waveform repeats after 1 second.

EFM waveform

The higher frequency must be at least twice the lower one. The higher frequency must also be sufficiently slow that the charging of the double layer does not contribute to the current response. Often, 10 Hz is a reasonable limit.

The current response is measured and processed. The result is a spectrum of current response as a function of frequency. The spectrum is called the

"intermodulation spectrum" (see example below). The two large peaks, with amplitudes of about 200 µA, are the response to the 2 Hz and 5 Hz excitation frequencies. Those peaks between 1 µA and 20 µA are the harmonics, sums, and differences of the two excitation frequencies. The Electrochemical Frequency Modulation analysis script uses these peaks to calculate the corrosion current and the Tafel constants. Between the peaks, the current response is exceedingly small. There is barely any response (<100 nA) at 4.5 Hz, for example.

EFM intermodilation spectrum

The frequencies and amplitudes of the peaks are not coincidences. They are direct consequences of the EFM theory.

Calculating the Corrosion Current

Taking the example above, a Base Frequency of 1 Hz is used together with Multiplier A = 2 and Multiplier B = 5. The table below shows the calculation of the frequencies used in the EFM calculation and the amplitudes expected. In the excitation waveform, 2 Hz and 5 Hz are present with equal amplitudes so it is not much of a surprise that the current response at these two frequencies should be equal.

Multiplier	Value	Frequency / Hz	Amplitude constant
A	2	2	a
B	5	5	a
B-A	5-2	3	b
B+A	5+2	7	b
B-2A	5-2·2	1	c
2B-A	2·5-2	8	c
B+2A	5+2·2	9	c
2B+A	2·5+2	12	c
2A	2·2	4	d
2B	2·5	10	d
3A	3·2	6	e
3B	3·5	15	e

The five constants a to e depend on the type of corrosion taking place. The equations below are valid if both the anodic and cathodic processes are activation- (kinetically) controlled. Similar equations can be derived for the case of passive metals or for a diffusion-controlled cathodic processes.

Amplitude constant	Value for Activation Control of Corrosion
a
a
b
b
c

where βa, βc = Tafel constants (in volts/loge unit; V = excitation amplitude (in volts). The value of the corrosion current icorr can be calculated from the measured amplitudes a, b, and c. Values of βa and βc can also be calculated. The corrosion rate can be obtained directly from the value of icorr.

Gamry tip bulb

The great strength of the Electrochemical Frequency Modulation technique is that, according to EFM theory, the ratio of the constant b to d should not depend on corrosion rate or Tafel constants. The ratio d/b is given the name "Causality Factor 2" because that ratio is 2.0! Likewise, the ratio e/c should be equal to 3.0 and is given the name "Causality Factor 3". These Causality Factors serve as an internal check on the validity of the EFM measurement!

Calculation of Corrosion Rate

When you fit corrosion data to a model, the numerical result is generally a corrosion current. We are interested in corrosion rates in more useful terms, such as a corrosion rate in millimeters per year. How is corrosion current used to generate a corrosion rate?

Assume an electrolytic dissolution reaction involving a chemical species, S:

You can relate current flow to mass via Faraday's Law.

where

•QS is the charge in coulombs resulting from the reaction of species S,

•n is the number of electrons transferred per molecule or atom of S,

•F is Faraday's constant = 96 485.34 coulombs/mole,

•MS is the number of moles of species S reacting.

A more useful form of Faraday’s Law requires the idea of equivalent weight. The equivalent weight (EWS) is the mass of species S that will react with one faraday of charge. For an atomic species, EW = AW/n (where AW is the atomic weight of the species). For a complex alloy that undergoes uniform dissolution, the equivalent weight is a weighted average of the equivalent weights of the alloy components. Mole fraction, not mass fraction, is used as the weighting factor. If the dissolution is not uniform, you may have to measure the corrosion products to calculate EW.

Substituting into Faraday’s Law we get:

where WS is the mass of species S that has reacted.

In cases where the corrosion occurs uniformly across a metal's surface, you can calculate the corrosion rate in units of distance per year. Be careful: this calculation underestimates the problem when localized corrosion occurs!

Conversion from a weight loss to a corrosion rate (CR) is straightforward. We need to know the density, d, and the sample area, A. Charge is given by Q = It, where t is the time in seconds and I is a current. We can substitute in the value of Faraday's constant. Modifying the previous equation,

where

•CR is the corrosion rate. Its units are given by the choice of K,

•Icorr is the corrosion current in amperes.

•K is constant that defines the units for the corrosion rate.

•EW is the equivalent weight in grams/equivalent

•d is the density in grams/cm³

•A is the area of the sample in cm².

The following table shows the value of K used in the corrosion rate equation for corrosion rates in the units of your choice.

Units for Corrosion Rate	K	Units
mm/year (mmpy)	3272	mm/(A cm year)
µm/year	3.272 · 106	µm/(A cm year)
mils/year (Gamry default) (mpy)	1.288 · 105	mils/(A cm year)

See ASTM Standard G 102, Standard Practice for Calculation of Corrosion Rates and Related Information from Electrochemical Measurements, for further information. The standard can be obtained from ASTM, 1916 Race St., Philadelphia, Pennsylvania 19013-1187 (USA) www.astm.org.

Current and Voltage Convention

A current value of –1.2 mA can mean different things to people in different areas of electrochemistry. To a corrosion scientist it represents 1.2 mA of cathodic current. To an analytical electrochemist it represents 1.2 mA of anodic current.

In the Electrochemical Frequency Modulation (EFM) software's default techniques we follow the corrosion convention for current:

•Positive currents are anodic, resulting in an oxidation at the metal specimen under test.

Potentials can also be a source of confusion. Throughout Gamry's software, the equilibrium potential assumed by the metal in the absence of electrical connections to the metal is called the open-circuit potential, Eoc. We have reserved the term corrosion potential, Ecorr, for the potential at which no current flows, as determined by a numerical fit of current-versus-potential data. In an ideal case, the values for Eoc and Ecorr are identical. One reason the two voltages may differ in real systems is changes in the electrode surface during the scan. In EFM, all potentials are specified or reported as the potential of the working electrode with respect to either the reference electrode or the open-circuit potential. The former is always labeled as "vs Eref" and the latter is labeled as "vs Eoc".

The equations used to convert from one form of potential to the other are:

Regardless of whether potentials are versus Eref or versus Eoc, one sign convention is used. The more positive a potential, the more anodic it is. More anodic potentials tend to accelerate oxidation of a metal specimen.

Literature

The following references are useful for learning more about the techniques that are available in DC Corrosion:

1.R.W. Bosch and W.F. Bogaerts, "Instantaneous Corrosion Rate Measurement with Small-Amplitude Potential Intermodulation Techniques", Corrosion, 52 (1996) 204.

2.R.W. Bosch, J. Hubrecht, W.F. Bogaerts, and B.C. Syrett, "Electrochemical Frequency Modulation: A New Electrochemical Technique for Online Corrosion Monitoring", Corrosion, 57 (2001) 60.