TAFELS’ LAW AND EXCEPTIONS.
In this section the behavior of single Redox reactions and combinations of multiple Redox reactions as a function of the potential of the system is investigated as well as exceptions to Tafels law. How Faradays’ law applies to corrosion for measuring corrosion rates will be outlined and laboratory testing will also be described.
Single Redox Reaction.
As noted above, if the relationship between the current flowing and potential for a reaction can be measured, then a straight line relationship should be found. A diagram showing typical Tafel behavior for a single reaction is shown in below.
Tafel lines for a single Redox reaction showing the effect on anode and cathode reaction rates as a function of polarization.
At equilibrium, a current is present but of very low magnitude. This is the exchange current density when the anodic current is matched by the cathodic current with no net flow external to the electrode. When the potential is raised to levels more positive than the equilibrium potential, then the Tafel lines are found to be followed. At potential more positive than the equilibrium value, the anodic Tafel line increases in current and the cathodic Tafel line decreases in current. The net reaction rate is the difference between the two currents from Faradays law discussed earlier. At more positive potentials the anodic reaction dominates and dissolution of atoms to ions occurs. Conversely, if the potential of the half cell is more negative than the equilibrium potential, then the cathodic reaction dominates. The shift of potential away from equilibrium values will then force a reaction either anodic or cathodic.
Two Redox Reactions.
The situation when Redox reactions are together is just an expansion of the single case. However, the rest potential of the system, or open circuit potential will fall in between the two equilibrium potentials as shown in figure 5. This has the effect of polarizing both reactions, one anodically and one cathodically. The value of the open circuit potential will be determined by the Tafel slopes for each reaction.
The overall effect of joining two different electrodes together is that one will corrode in an anodic reaction while the second electrode will get heavier as metal ions are deposited on the surface in the cathodic reaction. The corrosion rate can be determined from icorr measurements and the potential of the reaction will be given by Ecorr.
Variations from Standard Conditions.
So far all the reactions have been at the standard conditions of one atmosphere of gas pressure, one normal solutions, pure metals and a temperature of 25oC.
These are conditions that rarely exist and corrections considering the specific conditions must be made. One correction method is by the application of the Nernst equation given below:-
E = Eo + (RT/zF) ln K
where E - electrode half cell potential at the conditions, Eo - is the Redox potential, K- is the rate constant for the particular reaction.
Determination of the rate constant for a reaction.
The rate constant, K, is determined from the reaction constituents and products.
The activity of a dissolved substance is defined as:-
a= Vapor pressure of substance above solution
Vapor pressure of substance in standard state
For an IDEAL solution
The interactions between components are such that a solute has a vapor pressure directly proportional to its concentration:-
p1 = p1o c1
where p1 - vapor pressure in solution, p10 - vapor pressure in standard state and c1 - concentration in moles.
For a NON-Ideal solution.
A non-ideal solution behaves in reactions as if it has more or less solute than it actually has. In this case:-
p1 = p1o a1
where a1 - activity.
Some examples.
(1) Manganese in solution in iron behaves ideally and follows Rauolt's that
a= c
(2) Iron deviates from linearity in solution with both copper and silicon. Fe and Cu show little affinity for each other - a positive deviation, while FeSi is a compound formed by a negative deviation or great affinity. These follow Henrys law that
a1 = 10 c1
where 10 is the activity coefficient.
Consider an electrochemical reaction:-
Zn + 2HCl = ZnCl2 + H2
Note that this reaction is both chemically and electrochemically balanced.
Anode:-Zn -> Zn2+ + 2e-
Cathode:-2H+ +2e- -> H2
The reaction rate constant K, for this reaction is given by:-
K = pressure of oxidized species
pressure of reduced species
K = pZn2+
pZn
but from above:-
pZn = pZno aZn and pZn2+ = pZn2+o aZn2+
However, a=1 for pure solids, pure gases and 1N ionic solutions
K = PZn2+ = pZn2+o a Zn2+
pZn pZno
but po is also standard state and regarded as 1.
Therefore,
K = aZn2+
This is the reaction rate constant that will be used in the Nernst equation.
E = Eo + (RT/zF) ln K
E = Eo + 2.3(RT/zF) log K
Note: - 2.3 RT/F = 0.059 V at 250C.
Example for copper in 0.1 N CuSO4 solution.
Reaction:- Cu = Cu2+ + 2e-
Eo Cu/CuSO4 = 0.34 V
E(0.1N) = 0.34 + (0.059 log 0.1)
z
= 0.34 + (0.03 x -1)
= 0.31 V.
Polarization and Corrosion.
A practical case for corrosion is when a metal is placed in salt water. The metal has its own Redox reaction available of:-
M = Mz+ + ze-
In solutions to represent the marine environment a 0.5N NaCl solution in distilled water is employed. It contains some dissolved oxygen. A second Redox reaction is then available which is:-
O2+ 2H2O + 4e- = 4(OH-)
This is called the reduction of oxygen to hydroxyl ion reaction. Its Redox potential(Er) is +0.401 on the hydrogen scale. This will be different from the metal potential as it is a different reaction..
The potentials measured in the laboratory are therefore representative of the mixed potential of these two redox reactions. However, assume here that the Redox potential of the metal ion is more negative than the Reox potenial of the reduction of oxygen reaction, which is the case for iron in seawater. The potential measured by connecting the saturated Calomel electrode to a piece of iron in a beaker of seawater will be more positive than the Redox potential for the metal but more negative than the Redox potential for the dissolved oxygen reaction. Therefore both reactions are polarized, the metal reaction anodically and the dissolved oxygen cathodically. The metal will try to form metal ions in anodic half cell reaction while the dissolved oxygen will react with water to form hydroxyl ions in a cathodic half cell reaction.
To determine the corrosion rate in these circumstances, a test called a Tafel extrapolation is conducted. It basically obeys Faradays Law which states that:-
Q = (zFW)/M
Q - Coulombs; n - number of electrons involved in half cell reactions; F - Faradays Constant (96,500 coulombs or A.s)
W - weight of electroactive species; M - Molecular weight
Faradays Law applies to both deposition and corrosion. So in some cases “W” is weight gain, as in electroplating using the cathodic half cell reaction for a metal ion depositing, but in corrosion “W” represents a weight loss and uses the anodic half cell reaction, producing electrons.
W = (QM)/zF
Equivalent weight (EW) = M/z
W = (Q x EW)/F
Q = it
i - current in Amps; t - time in seconds
W= itEW/F
W/t is the corrosion rate in gm per sec.
W =dV where d- density and V - volume
dV = itEW/F
V = TA where T -thickness and A is the surface area
T/t= iEW/(FdA)
T/t is mils per year when appropriate constant are imposed for the number of seconds in a year and for the unit changes for length.
Corrosion rate in mils per year is then
CR = i x EW x 31.6 e9
d x F x A x 2.5 e6
i/a is the current density in A/cm2
CR = 0.13 (icorr x EW)
d
for units of Corr Rate in mils per year, icorr in A.cm-2
EW is the equivalent weight and d is the density of the metal.
Problem:-
Calculate the corrosion rate in mils. per year from the following data:-
MetalIcorr(A.cm-2)
Fe1000
Fe0.1
Ti0.03
Zr0.01
Electrochemical Laboratory Testing.
For electrochemical measurements a three electrode system is used. The test piece is usually the working electrode. It is the electrode under investigation, normally to determine its anodic behavior but sometimes to determine its cathodic behavior. This is connected through a voltmeter to the reference electrode, usually a Saturated Calomel electrode. Both the working electrode and the SCE are in a flask containing the electrolyte. Usually this electrolyte will contain dissolved oxygen. To try and provide similar conditions from test to test and lab to lab, a test protocol exits for localized corrosion testing as ASTM G5. This specifies that the solution should be purged with nitrogen gas for one hour prior to the test. This displaces some oxygen and brings it to a constant level of around 8 parts per million. The voltage measured is therefore the difference between the SCE and the potential at which the reduction of oxygen as a cathodic reaction and the anodic reaction on the working electrode surface. For 4340 steel in 0.5N NaCl this is around -0.625 V. The steel is uniformly corroding at this potential. The potential measured is called the open circuit potential or Eoc. To convert this to Standard Hydrogen Electrode scale, it should be remembered that the SCE scale is +0.242 V on the SHE scale. The Eoc(SHE) is therefore -0.383 V. Note that this is more positive than the Eo for the Fe/Fe2+ reaction of -0.44 V and indicates the iron is anodic and hence corroding.
The third electrode connected to the other two is the counter electrode. This electrode is used to provide a current flow to the working electrode for either anodic or cathodic conditions. The current through this electrode is measured by a sensitive ammeter. The counter electrode is usually made of a very noble material to avoid any problems with its dissolution.
The equipment used to control and interconnect the three electrodes is called a Potentiostat. A potentiostat controls the potential between the working electrode and the reference electrode while simultaneously measuring the current flowing into or out of the working electrode necessary to maintain the selected potential.
A typical laboratory test is a potentiodynamic scan. The potential is ramped at a fixed rate per second in a potentiostat and the current response of the working electrode measured. The potential is then plotted as a function of the log of the current density to produce polarization curves or Evans diagrams. The test conditions are identified in the ASTM standards. A slow scan rate is desirable to enable a quasi equilibrium state, especially if passive metals are involved. Typical data is shown above for a potentiodynamic scan. From the data, back extrapolation of the straight line portion of the anodic and cathodic semi logarithmic behavior enables the calculation of the current density at open circuit for input into the Faraday equation to calculate corrosion rate.
Variations on Tafel Behavior.
Not all materials exhibit Tafel behavior over a wide potential range. Variations in the Tafel law behavior can be split into two types. The first is when the reactive species cannot transport either away from or to the electrode fast enough to maintain the reaction rate. This process is called concentration polarization as it depends on concentration reaction control. The second is due to film formation on the electrode during anodic polarization in a process called passivation.
Concentration Polarization.
The figure below shows the behavior when concentration polarization is occurring for a cathode reaction.
Figure 6 shows the behavior associated with concentration polarization.
The specific reaction in this case if for the reduction of oxygen in water to form hydroxyl ions.
a is the activation polarization from Tafel law and is controlled by the activation energy for the reaction. c is the concentration polarization due to reaction being limited by the transport of components. The current density of this reaction is ilim of limiting current density. Its value can be calculated to be:-
ilim = (D z F Co)/x
where D - diffusion rate of the species, Co - concentration and x - diffusion distance. One of the best known concentration limiting reactions is the reduction of dissolved oxygen in water. The maximum limiting current density of this reaction is around 150 A/cm2, due to the low solubility of oxygen in water. However lower values than this can be found if the surface does not support the reaction. This is useful as it often limits the corrosion rate of very anodic metals. A similar upper limit is found for laboratory experiments when anodic polarization of steels in 0.5N NaCl reaches a limiting current around 103A/cm2. In this case the ions cannot be transported away from the anode quickly enough.
In the figure above, the vertical line is the limiting current density for the cathodic reaction for dissolved oxygen in water.
The second major variation on Tafel behavior is by film formation or passivation. This behavior is shown in figure 7. In this case the anodic reaction is such that a protective film is formed on the surface of the electrode. Note the current density for this process is low so only a small active region is necessary for supplying the ions incorporated into the anodic film. Alloy development of stainless steels is based on the passivation process. An example of industrial application of these films is anodizing.
icrit is the critical current density for film formation to supply the metallic ions to be incorporated into the film. Epp is the primary passivation potential for the film. Both these parameters must be exceeded for stable film formation. The most useful films are those with low primary passivation potentials and low critical current densities.
The transpassive region is the potential at which the dissolved oxygen in water reaction transforms from cathodic to anodic and the films starts to break down. On important exception to the transpassive behavior is titanium which in water does not exhibit the transpassive behavior.
In the figure above two sets of data have passive ranges, while one does not..
Applications of the Nernst Equation.
A considerable effort has gone into the production of diagrams called Pourbaix diagrams named for the person who developed them. These diagrams map out regions of immunity, corrosion and passivity for metal systems by applying the Nernst equations to electrochemical reactions.
Pourbaix Diagrams.
Pourbaix, a French scientist used the Nernst equation to predict the relationship between potential and the solution pH to predict whether an electrode would be immune, active or passive in the environment(1). These diagrams are therefore maps of the state of an electrode as a function of both potential and pH. They were constructed from thermodynamic data and are available for pure metals. The kinetic data used by Pourbaix was the solubility product of the reactions. If this value is low, then a stable film is formed in the environment. If the value is high then the film dissolves and is non protective. Examples of the formation of a Pourbaix diagram and its use will be provided in the following section.
Following the general reaction for electrodes of:-
M= Mz+ + ze-
This reaction is pH independent as it does not involve H+ or OH- ions in the reaction. The potential pH behavior can then be determined from application of the Nernst equation covered in the previous section.
EM/Mz+ = Eo + 2.3 RT log aMz+
zF
Pourbaix chose an ionic activity of 10-6 moles/liter for the active species.
Eo = -0.44V for Fe/Fe2+
aMz+ = 10-6
EM/Mz+ = -0.44 + (0.06 x -6)
2
= -0.44 - 0.18
= -0.62 V.
Using the same conventions as before this indicates that at a potential more positive than -0.62V iron will be active and corrode. At potentials more negative than -0.62V the iron will be immune and not corrode as the reaction favored is the deposition of atomic iron from the ionic species. This allows a horizontal line to be drawn on a plot of potential on the vertical axis and pH along the horizontal axis.
Other reactions should be considered:-
xMz+ + yH2O = MxOy + 2yH+
For the iron case, this reaction represents iron in solution reacting with water to form a metal oxide releasing hydrogen ions. This reaction is pH dependent as hydrogen ions are involved. It is not potential dependent as no electron transfer is involved
Fe2+ + 2H2O = Fe(OH)2 + 2H+
Fe(OH)2 solubility is 10-14.7 moles/liter: 10-6 ions of Fe in solution – Pourbaix assumption
We need to find the relationship between ion concentration and pH.
[H+][OH-] = 10-14
log H+ + log OH- = -14
-log H+ = pH
log OH- = pH – 14
Rewrite above equation by dissociating water:-
Fe2+ + 2OH- +2H+ = Fe(OH)2 +2H+
removing the 2H+ on either side can be written as:-
Fe2+ + 2OH- = Fe(OH)2
K = [Fe2+] [OH-]2
[Fe(OH)2]
Oxidized species is the highest valence positive ionic form or side where electrons are produced
As Fe(OH)2 is solid its activity is 1
log[Fe2+] + 2log[OH-] = -14.71
Substitute for [OH-] from above
logFe2+ + 2(pH-14) = -14.71
At Pourbaix conditions of Fe2+ = 10-6
-6 + 2pH -28 = -14.71
pH = 34 -14.71
2
pH= 9.64
For the above reaction a vertical line is drawn on the potential pH diagram. At pH greater than 9.64, iron does not corrode actively as any ions formed in a Tafel type reaction are immediately consumed in forming the hydroxide of iron as a stable non dissolving film or a passive film. One use of this reaction is reinforcing bars in concrete. Concrete has a pH of 12.5 and so it places steel in the center of the passive zone.
Another possible reaction is:-
2Fe2+ + 3H2O = Fe2O3 + 6H+ + 2e-
Note that both hydrogen ions and electrons are involved in this reaction. This indicates it will be both pH and potential dependent. As there is plenty of water, aH2O = 1
K = [Fe2O3] [H+]6
[Fe2+]2
aFe2O3 = 1 as it is in solid form.
log K = 6logH+
2logFe2+
log K = 6logH+ - 2log Fe2+
Nernst equation ;-
EM/Mz+ = Eo + 2.3 RT log K
zF
= Eo + 0.06 (6log H+ - 2logFe2+)
2
= Eo -0.18pH-0.06log Fe2+
aFe2+ = 10-6,logFe2+ = -6
= Eo-0.018pH+0.36
Eo for the Fe2+/Fe3+ = 0.771
E= 0.771 + 0.36 - 0.18pH
E= 1.131 - 0.18pH
This reaction is therefore both pH and potential dependent and will be a sloping line on the Pourbaix diagram.
This indicates the method of mapping the space on a Pourbaix diagram. The iron diagram is shown below.
The diagram for chromium is also shown (1). They indicate that zinc by and large protects by being active at pH values from 5 to 7 while chromium is passive over a similar region
due to passive film formation.
There are two other important reactions that need to be superimposed on the Pourbaix diagram, reduction of oxygen and reduction of hydrogen.
O2 + 2H2O + 4e- = 4(OH-)
More noble potential than this value, oxygen if formed as the reaction goes from hydroxyl ions to oxygen. At potentials more negative than this value, dissolved oxygen is reduced to hydroxyl ions. As solution conditions change the Nernst equation must be applied.