Aqueous Corrosion

15.1  Introduction
Metallic iron was first produced about 2000 years ago and has been corroding ever since. One type of  corrosion is reaction of a metal with water that converts the surface to an oxide or hydroxide by reactions such as:

M + z H2O -->MOz/2 + zH2        or         M + zH2O -->M(OH)z + zH2          (15.1)

For most metals, the oxide or hydroxide is thermodynamically more stable than the elemental metal. This is the case for iron, nickel, chromium, aluminum, zinc, uranium, and very importantly for this book, zirconium. It is not so for the noble metals, notably gold and platinum.

When a metal corrodes, the corrosion product behaves in one of several ways: i) the corrosion product may flake off the metal's surface, exposing fresh metal for continuation of the process; this is the case for pure iron and zirconium alloys; ii) the corrosion product may adhere to the metal but continue to grow, as with ????; iii) a very thin corrosion-product layer may adhere to the surface of the metal but act protectively and greatly slow the corrosion rate. This phenomenon is called passivity and fortunately for modern society, applies to aluminum and chromium and their alloys with other metals. Corrosion is said to be uniform if the product layer is roughly the same thickness over the entire metal surface. If only special spots on the metal are susceptible to oxidation, corrosion is termed localized. The latter form of corrosion may respond to a tensile stress parallel to the metal surface.

Alternatively, the metal may corrode by effectively dissolving in the aqueous solution by a reaction such as:
M + zH+ --> Mz+ + zH2                                                   (15.2)

This mode of corrosion, called active corrosion, is particularly common in highly-acidic solutions. It is not a practical concern since the process is fast and such a combination would render useless the component in such an environment.  However, the basic theory of corrosion is developed in Sects 15.2 - 15.6 based on this type.

 In reactions (15.1) and  (15.2), z designates the valence or oxidation state of the metal. Some metals, aluminum being one, have only a single oxidation state. Others such as iron have multiple oxidation states, all or some of which may be present as solid oxides or hydroxides coating the metal or as ions dissolved in the aqueous phase.

This chapter explores the thermodynamics, kinetics and mechanical aspects of corrosion.
The thermodynamic aspects of corrosion are twofold: First, it determines whether a particular metal is susceptible to corrosive attack by a particular aqueous solution. Second, if a metal is susceptible to corrosion, approximate thermodynamically-based constructions called Pourbaix diagrams show the range of conditions under which corrosion does or does not occur and which of  reactions (15.1) and (15.2) is the route. These diagrams recognize the dependence of the corrosion process on the pH and the electric potential of the aqueous phase (see Sect 2.9.1).

The above corrosion processes are called overall reactions. They do not explicitly show the transfer of electrons from one element to another. To do so requires splitting them up into                  half-cell reactions, which clearly show the electron transfer process. As discussed in Sect. 2.9, for reaction (15.2), these are:
M = Mz+ + ze-             2H+ + 2e- = H2                                               (15.3)

where e- denotes an electron. These electrons are not found floating around in the solution like the ions; they are transferred directly from one species to another. In reaction (15.3), each metal atom M donates z electrons to z hydrogen ions to effect the conversion to aqueous ions of Mz+ and hydrogen gas.
The --> signs in reactions (15.1) and (15.2) indicate that the reaction is proceeding from left to right. The equal signs in reaction (15.3) signify that both the forward and reverse processes occur, and moreover, that they are of equal speed. That is, the reaction is in equilibrium. This is not always true as half-cell reactions can proceed at finite rates. This kinetic aspect of corrosion is expressed by Tafel diagrams. Analysis of the kinetic mechanisms can explain the origin of these diagrams.

Localized corrosion is attack of the metal in penetrations called pits or crevices. The former look like holes dug in the ground, and can often be seen on stainless steel tableware. Crevices, on the other hand, resemble clean cracks. Crevice corrosion most frequently occurs in metals under stress, which is the origin of the name stress-corrosion cracking (SCC). This phenomenon is sensitive to the pH, the electric potential and the presence of other ions in the aqueous solution. In certain alloys, chloride ions in solution significantly affect SCC. Cracks can penetrate a specimen either through the grains of the metal or follow grain boundaries. The latter is most common and is called intergranular stress-corrosion cracking, which is shortened to IGSCC. A special variant of IGSCC occurs in the coolant of nuclear reactors, in which neutron or gamma irradiation creates a variety of radiolysis products that significantly accelerate the corrosion process. The term for this effect is irradiation-assisted stress-corrosion cracking, or IASCC.

The thermodynamics and kinetics of corrosion reactions depend on temperature. Because the liquid phase is water, 25OC is the most common temperature considered. The notable exception is the water coolant of a nuclear reactor, which is within ±20oC of 300oC.

15.2  Thermodynamics of corrosion

15.2.1 The overall reaction
Whether corrosion of a metal is thermodynamically possible depends on the free-energy change of the overall reaction. For the first of reaction (15.1), for example, this is:
(15.4a)
where the superscript o indicates that the substance is in its standard state, which is pure, at the temperature of the reaction, and at 1 atm pressure. The two solids and water are pure (or in the latter case, nearly pure) and so are in their standard states. Hydrogen gas may be present at a pressure different from 1 atm, which accounts for the second term in parentheses in Eq (15.4a).
Equation (15.4a) can be written as:                  + RTln                                  (15.4b) where is the standard-state free-energy change of the reaction. It depends on temperature
only. Reactions take place if Dg is negative, which almost always means that is negative. For example, if M = Fe and z = 2, in the first of reactions (15.1), Dgo = -7.7 kJ/mole at 25oC. Forming the equilibrium constant of the reaction as:
K = = exp(-Dgo/RT) = exp[-(-7700)/8.314x298] = 460 atm
means that the hydrogen pressure must be greater than 460 atm in order to prevent all Fe from becoming oxidized to FeO in water. On the other hand, if M = Pb, Dgo = 20 kJ/mole and lead contacting water could not be converted to PbO unless the H2 pressure were less than 3x10-8 atm.
The standard-state free energy simply indicates whether the left-hand side or the right-hand side of the written reaction, of which (15.1) and (15.2) are examples, is favored.

15.2.2  Pourbaix diagrams
More detailed information about the thermodynamics of a particular metal in water is contained in Pourbaix diagrams, named after the Russian chemist who, while studying in Belgium, produced a PhD thesis on the graphical representation of the role of the pH and electric potential of an aqueous solution on the stable states of the contained metal. Pourbaix diagrams are also known as pH-potential diagrams. They are analogous to binary phase diagrams (Sect. 2.7), which are also called temperature-composition diagrams.

 Pourbaix diagrams are constructed from Nernst equations of Eq (2.77a). For the M/Mz+ half cell at 25oC, this is:
(15.5)                     where fo is the standard electrode potential (Table 2.2), z is the number of electrons transferred in the half-cell reaction and [Mz+] is the molar concentration of the metal ion. Solid M does not appear since is activity is unity because it is assumed to be pure.

The rules for using other Nernst equations to construct Pourbaix diagrams are:
i) the pressures of the gases H2 and O2 (if they appear in the equation) are set equal to 1 atm                 ii) the concentrations of all ions (excepting H+) are fixed at 10-6 M. 
The latter condition is intended to separate regions in which the individual species predominate. As in the Nernst equations, the anion that accompany the various cations (e.g., Cl-, SO ) are ignored, although the diagram is slightly dependent on these species.
b 
The two states of the metal in Pourbaix diagrams are dissolved (as ions) or solid, including metal: metal oxide(s), hydroxides and oxy-hydroxides. For iron, the important species are the ions Fe2+, Fe3+, Fe(OH)2+, Fe(OH)  and FeO and the solids Fe, Fe3O4 and Fe2O3.

To account for the different sign convention used in Table 2.2 and Eq (2.77a) from the convention used in corrosion analyses, the Nernst potential in Sect. 2.9.3 (f) is the negative of equilibrium electrode potential, eeq  in the present notation:

eeq = -f (Sect. 2.9)                                         (15.5a)

An example of constructing the  Pourbaix for iron is shown in Table 15.1.
The two horizontal lines (nos. 1 and 2) and the vertical line (no. 4) and the sloped line (no.3) are shown on the Fe/H2O Pourbaix diagram of Fig. 15.1 along with the remainder of the iron-water lines.
Table 15.1  Construction of the iron Pourbaix diagram at 25oC                                                                 

(molar concentrations are denoted by [  ])

Line No.	couple	Half-cell reaction	Standard potential, V@	Log (ion concentration ratio)	Pourbaix line
1	Fe/Fe2+	Fe = Fe2+ + 2e-	0.44	log[Fe2+] = log[10-6]	= - 0.62
2	Fe2+/Fe3+	Fe2+ = Fe3+ + e-	-0.77	log[Fe3+]/[Fe2+] = 0	= +0.77
3	Fe2+/Fe2O3	2Fe2+ + 3H2O = Fe2O3 + 6H+ + 2e-	-0.68*	log[H+]6/ [Fe2+]2 = - 6pH - 2log[10-6]	= 1.03 - 0.18pH
4	Fe3+/Fe2O3	2Fe3+ + 3H2O = Fe2O3 + 6H+	#	log[H+]6/ [Fe3+]2 = - 6pH - 2log[10-6]	pH=1.52#
a	H2/H2O	2OH- + H2 =                     2H2O + 2e	0.83	log(pH2/[OH-]2) = 2log10-14 + 2pH	= - 0.059pH
b	O2/H2O	H2O = 1/2O2 +      2H+ + 2e	-1.23	log( /[H+]2) = 2pH	= 1.23 - 0.059pH

@ From Table 2.2
* see example in Sect. 2.9.2

#  line No. 4 does not involve electrons, so must be treated like reaction (b) in the example in Sect. 2.9.2:
In Table 2.2, half-cell reactions (15) - 2´(9) give reaction No. 4 in the above table.  Using Eq (2.75):

Equating the right-hand sides of the above equations:   -5.75(-6pH + 12) = -16.6. Solving: pH = 1.52.
 

The lines a and b in Fig. 15.1 encompass the zone of stability of water. For potentials less than line a, water decomposes into H2 and OH- by the reverse of half-cell reaction 3a in Table 2.2. The standard electrode potential of this half-cell reaction is = 0.83 V (see problem 2.1).  Its Nernst equation (with 10-6 M concentrations for ions and 1 atm pressure for gases) is obtained from:

Because of this restriction, metallic iron is not stable in water under any accessible conditions.
For potentials greater than line b in Fig. 15.1, water decomposes into O2 and H+ by half-cell reaction 1 in Table 2.2, for which the standard electrode potential is  = - 1.23 V . This leads to the Nernst equation shown in the last row of the last column in the above table (with the sign change to conform to the convention).  

Fig. 15.1 Pourbaix diagram for iron/water at 25oC

15.3  Electrostatics - the electric potential and the electric field

The electric potential plays a central role in corrosion. It is analogous to the chemical potential, the thermal potential and the mechanical potential; all of these have in common that a gradient of the potential causes motion of some quantity: the chemical potential drives material species; the thermal potential drives heat; the mechanical potential drives objects. The electric potential drives charged species, either electrons in a wire or ions in an aqueous solution.
Consider two point charges, q and q' separated by a distance r in a some medium.

 Charge q is fixed in space but charge q' is moveable. The force acting on q' in the positive r direction (to the right in the diagram) due to the presence of q is:
(15.6)
eo is the vacuum permittivity, 8.85´10-14 C/Volt-cm and Kr is the dialectic constant of the medium. For a nonpolar medium such as liquid helium, Kr = 1, meaning that the force in this medium is the same as .if the two charges were in vacuum. In general, Kr is a measure of how much the force is reduced by the presence of the medium. The magnitude of Kr is ³ 1, by how much depends on the dipole moment of the molecules in the medium. For polar molecules such as water,  positive and negative charges each of magnitude s are separated by a distance d, the dipole moment is sd. Most molecules have dipole moments somewhat greater than  unity. Very polar molecules such as HF and H2O have large dipole moments, and as a result, their dielectric constants are ~ 30 and 80, respectively. By aligning their dipoles in the r-direction they effectively reduce the force F acting between the two charges.

If the charges q and q' are of the same sign, the force on q' is in the +r direction, as  shown in the above diagram. The work done in bringing q' from infinity to r is:
(15.7)
The negative sign in front of the integral is because the charge q' is moved in the -r direction.

If q' = +1, the force is called the electric field:
(15.6a)
and the work is the electrostatic (or electric) potential:
(15.8)
The units of f are volts and those of E are volts/m. q in Eqs (15.6a) and (15.8) need not be a point charge; in general it is a distribution of charges such as those that gather at the surface of a metal immersed in a solution containing positive and negative ions.

The variation of electrostatic potential (f) across an interface for the case of excess negative charge on the aqueous solution side of the interface and excess positive charge in the metal is shown on the left in Fig 15.2.  The right-hand sketch shows the variation of electric field across the interface. 

Example: What is the potential difference between two hollow, initially-neutral metal spheres each 2 cm diameter after 125 mC of electrons have been moved from one sphere to the other?  The center-to-center separation of the two spheres is 10 cm. The capacitance of the two-sphere system is ~6´10-13 Coulombs/Volt. The capacitance (cap) is defined as cap = q/Df, where the charges are +q on one sphere and -q on the other. For this situation, q = 125 mC, so Df = q/cap = 125´10-6/6´10-13 = 2´108 Volts.

15.4  Driving force for corrosion - the interface potential difference

15.4.1 The electric double layer
When a metal is inserted into an aqueous solution containing anions and cations, a redistribution of charges at the metal-solution interface takes place. Figure 15.3 shows a representation of this effect. In this example, the metal is assumed to be negatively-charged with respect to the solution. The inner layer in the solution shows water molecules lined up at the metal surface. The outer solution layer shows solvated cations, anions and randomly-oriented water molecules. No attempt has been made to show hydrogen bonding of the water molecules. Because the metal at the interface is negative, the solution at the interface  has an excess of positive ions.  The bulk metal and the bulk solution are electrically neutral. The situation shown in Fig. 15.3 could

Fig. 15.2  Electric properties of a metal/solution interface

have been reversed, with the metal positive and the solution negative (excess of anions in the layer). One can imagine separating the metal and the solution without disturbing the charge
distributions, then measuring the electric potential of each surface by recording the work required to move a positive single charge from far away towards each of the surfaces. This experiment is not possible, so the difference in the potentials across the interface,

Df = fM - fS                                                               (15.9)
cannot be measured.

15.4.2  The electrode potential at equilibrium
The electrochemical cell shown in Fig. 15.4 provides information on the equilibrium electrode potential, . For fixed concentrations of Mz+ in the solution (and no current flow), the measured cell potential contains other contributions:
(15.10)
where "other" includes potential changes in the measurement apparatus. The cell in Fig. 15.4 is similar to that in Fig. 2.9. The M/Mz+ half-cell reaction does not depend on the pH and the H2/H+ half-cell reaction does not involve Mz+, so the cells in Figs. 2.9 and 15.4 measure the same thing.

15.4.3 Electrode potential during corrosion
In analyzing corrosion kinetics, the quantity appearing in the rate equations (Sect. 15.5.2) is the difference , where e is the electrode potential for metal M and a solution wherein the rates of the reverse and forward reactions of are not equal (i.e., a corrosion current flows).  For the purpose of measuring e, a somewhat more complex setup, such as the one shown in Fig 15.5, is needed. Here the voltmeter measures the electrode potential e while corrosion is occurring in the test electrode M.  The right-hand side of the system establishes a current i that is measured by an ammeter. This represents the corrosion rate, which is a function of the current and the concentration of Mz+.

Fig. 15.4  Electrochemical cell to measure eeq of the M/Mz+ half cell
If the metal half-cell reactions are pH-sensitive, a bridge (shown) is needed to allow separate solution compositions around the Pt electrode to the left and the M/Pt cell to the right.

15.5  Corrosion kinetics - the Butler-Volmer equation

The active corrosion process consists of two half-cell reactions: anodic (oxidation), M-->Mz+ + ze- and cathodic (reduction),  Mz+ + ze --> M. The difference between the rates of these two half-cell reactions is the overall oxidation rate:                                           
(15.11)

j is the flux of the species moving to and from the metal surface and the bulk solution. The units of j are moles of ions  per unit time per unit area. The right-pointing arrow above the j indicates the metal-to-solution direction and the subscript a means anodic. Similarly, the left-pointing arrow signifies transfer of the ion from the solution to the metal, which is a cathodic reaction indicated by the subscript c.

15.5.1  Microscopic effect of the interface potential
For a atoms in a metal to be converted to the ion in solution in the absence of an interface potential, not only must the standard free-energy change of the half-cell reaction be negative (the thermodynamic requirement) but the activation energy of the reaction (the kinetic requirement) must be accounted for. The solid curve in Fig. 15.6 shows  the changes in free energy of a metal atom as it is removed from the solid, converted to an ion and transferred to the bulk solution. In the diagram, Dgo is the negative of the standard-state free energy change for M --> Mz+ + ze- and Dg* is the activation energy. Df is the potential change across the double layer at the metal/solution interface (Sect. 15.4.1).

Fig. 15.6  free energy barrier for M --> Mz+ + 2e- in the absence of electrical effects (solid curve) and with the metal positivelycharged (dashed curve).                                                                        (The extent of the curves is about 10 nm.)

15.5.2  Rate equations
The rate (flux) of the M --> Mz+ + ze-  half-cell reaction is:

is the pre-exponential factor of the rate constant for this step. The activation energy is due to the energy barrier presented by the solid curve in Fig. 15.6.

The dashed curve in Fig. 15.6 shows the reaction barrier with the interface potential difference of Sect. 15.4 taken into account. This curve starts out at the metal surface with a potential (now electrochemical) at a free energy zFDf higher than the electrically-neutral value of Dgo (see                       Eq (2.71)) This added potential decreases with distance into the solution so that after a few nanometers, none of it remains. The dashed and solid curves at the maximum differ from the value at the interface by a fraction 1-a, where a < 1. The activation energy for the forward step with the interface potential is obtained from the plot as:
zFDf + e* = Dg* + zF(1-a)Df, or  e* = Dg* - azFDf
At this point, Df in the above equation is replaced by eeq . Although according to Eq (15.9a), these two measures of the potential differ by an unknown constant, the term this introduces into the rate equation is simply absorbed into the unknown rate constant . With this change, the anodic rate becomes:
(15.12a)

The rate of the cathodic step, Mz+ + 2e- --> M, in the absence of an electric potential is given by:

Including the interface potential, Fig. 15.6 gives the new activation energy as                                            Dgo + Dg* + (1-a)zFDf. Again replacing Df with eeq  gives the interface-potential-modified cathodic rate:
exp{-(1-a)zFeeq }                            (15.12b)

In this electrochemical equilibrium situation, the anodic and cathodic  rates are equal to each other and are called the equilibrium rate because there is no net flux: jeq = ja = jc , or:

 =   exp{-(1-a)zFeeq }       
(15.13)
With an applied potential e ¹ eeq, the net oxidation rate is .and eeq in Eqs (15.12a) and (15.12b) is replaced by e.  Dividing Eq (15.12a) by the first equality in Eq (15.13) and                   Eq (15.12b) by the second equality yields the Butler-Volmer equation:

                               (15.14)
where h is the overpotential:
(15.15)
When h= 0, there is no net consumption of metal, or jox = 0.  Given the corrosion conditions (the overpotential h and the cation concentration [Mz+] to fix eeq ), the net or oxidation rate jox can be calculated from Eq (15.14) if jeq is known.

15.6 Tafel Diagrams
From here on reaction rates are expressed in terms of current density instead of the flux:
i = zFj                                                            (15.16)
Both the rate j and the current i are per unit area. In this application, Faraday's constant is 96,500 Coulombs/mole.

15.6.1  The exchange current density
jeq corresponds to the exchange current density io = zFjeq.  Observing the sign convention in using the Nernst equation, the equilibrium electrode potential from Eq (2.77a) for the M/Mz+ half cell is:
(15.18)
The first equality of  Eq (15.13) becomes:
(15.18a)
and the second equality gives:                                                                                                                   
exp[-(1-a)zF(-fo)/RT][Mz+]a                           (15.18b)
Both of the above equations are of the form
ioM = QM[Mz+]a                                                         
where QM is the collection of constants, which, for these two expressions for ioM, must be equal. (this simply provides a relation between , which is not of interest). What is important is the dependence of ioM on the concentration of metal ions in solution raised to a power a ~ 0.5.             

For the hydrogen half-cell reaction, H2 = 2H+ + 2e

the exchange current density is:
ioH = QH[H+]a[pH2] (1-a)                                             (15.19)

In neutral water with the H2 pressure ~ 1 atm, the exchange current densities are:

The exchange-current-density concept is applicable to any combination of half-cell reaction and electrode. In general,
io = Q[oxid]a[red](1-a)                                               (15.19a)

where oxid and red are the ions in solutions or the gases taking part in the half-cell reaction:

red = oxid + ze-
15.6.2 Tafel diagram for half-cell reactions
A Tafel diagram is a plot of the Butler-Volmer equation.  For the half-cell reaction                                             M « M2+ + 2e- with the following numerical values:
a = 0.5;   z = 2;   F/RT  = 96.5/8.314´0.298= 39

and replacing the fluxes (j) with the current densities (i), Eq (15.14) becomes:
(15.14a)
Figure 15.7 is a plot of Eq (15.14a) with the alteration of changing the ratio i/io to its absolute value. This is needed because at negative overpotentials, the current changes from anodic to cathodic, so the second term in Eq (15.14a) dominates the first. Because negative values cannot be plotted on a logarithmic scale, the absolute value of the cathodic current is reported.

At large absolute values of the overpotential, either the first or the second term in Eq (15.14a) predominates. The slopes of the two branches are equal in Fig. 15.7 because a = 0.5. This is not generally the case, and for a ¹ 0.5, the slopes of the two branches differ.

The curvature of the two branches at small overpotentials is due to the two terms in Eq (15.14a) having the same order of magnitude. As h --> 0, the current density i approaches zero. However, for large overpotentials, only one of the two terms is important. The intercept of the extrapolation of dominant term at h = 0  is i/io = 1 (dotted lines in Fig. 15.7).

 

Fig. 15.7  Generic Tafel diagram for active corrosion

The Tafel diagram can be determined experimentally with the electrochemical cell shown in Fig.15.5. Provided by this apparatus is the current density (i) flowing from the  metal electrode M to the right-hand Pt electrode where the cathodic half-cell reaction 2H+ +2e- --> H2 takes place. The current density can be changed with the variable resistor in the diagram.  For each current density setting, the standard hydrogen electrode (SHE) on the left in the diagram measures the electrode potential. When the current is zero, the SHE measures the equilibrium potential. The overpotential (h) is then determined by Eq (15.15). The data from this experiment is sufficient to extract the parameters a and io.

Example   The plot below shows the data from the apparatus  of Fig. 15.5 for the half-cell Fe = Fe2+ + 2e- in acid of pH 2 at 25OC. What are the best-fitting values of a and io?

For this half-cell, z = 2 and the hydrogen ion concentration does not enter.
The extrapolations of the single-exponential portions of the plots (dashed portions) intersect zero overpotential at a current density of 1 mA/cm2, so this is io.
The slope of the linear portion of the anodic curve is log(i/io)/h = 1.86/0.1 = 18.6. From Eq (15.4) with jox/jeq replaced by i/io, the slope is:
,   or   

The slope of the cathodic branch is -log(i/io)/h = 1.90/(-0.126) = -15.1. But:
or       
Agreement between the values of a from the anodic and cathodic polarization curves is excellent.

 

15.6.3 Tafel diagram for overall cell reactions
The previous section dealt with individual half-cell reactions, which were caused to proceed in either the anodic or cathodic direction by application of the appropriate overpotential (positive or negative) to the electrochemical apparatus shown in Fig. 15.5. However, in actual corrosion situations, a metal (e.g., a pipe or tubing through which water flows) supports the anodic reaction M --> Mz+ + z e- along with a cathodic reaction, which can be the reverse of any one of the first four half-cell reactions in Table 2.2.  The cathodic half-cell reaction 2H+ + 2e- --> H2 will be employed for purposes of illustrating how both half-cell reactions operate together spontaneously.

Figure 15.8 is a schematic of water flowing through a pipe. The water is sufficiently acidic that the metal of the pipe wall undergoes active corrosion (i.e. without formation of an oxide scale on the surface). The anodic and cathodic half-cell reactions take place on different parts of the pipe inside surface. Electrons produced by the anodic reaction flow through the pipe wall to the zone where the cathodic reaction is occurring. One section of the pipe is connected to a standard hydrogen electrode (SHE) for the purpose of measuring the overpotential (i.e., the pipe is corroding). The pH 1 acid of the SHE is connected to the pipe water by a salt bridge, the function of which is to complete the electrical circuit of the measuring cell.

Fig. 15.8  Corrosion of a pipe in flowing water.

The Tafel diagram for this overall-cell process is shown schematically in Fig. 15.9.  It is based on simplified forms of the Butler-Volmer equation of Eq (15.14) for large overpotentials where only the first exponential term is needed for the anodic line and the second exponential term applies to the cathodic reaction. The lines in the diagrams are called polarization curves, although the curvature does not appear in the simplified diagrams of Figs. 15.9, 15.10 and 15.11.

The Tafel diagrams in Fig. 15.9 differ from the half-cell diagram of Fig. 15.7 in the definition of the overpotential. In Fig. 15.9, two half-cell polarization curves (lines) are superposed, each with its own overpotential defined by h of Eq 15.15. However, the equilibrium potential eeq for the cathodic and anodic reactions are different, yet the reference potential must be the same for both half-cells in order to properly use them together.

In Fig. 15.9, the reference potential has been chosen as the equilibrium electrode potential of the anode metal M, . Ignoring the curvature near the equilibrium potential seen in Fig 15.6, the anodic line for the metal is given by:
(15.19a)
where ioM is the exchange current density for the anodic half-cell reaction M --> Mz+ + ze- at a metal ion concentration of [Mz+] from Eq (15.18a). The parameter bM is:

                                                          (15.20a)

When coupling the anodic line of the metal diagram to the cathodic portion of the hydrogen diagram,  the reference potential must be the same for both diagrams. The overpotential is with respect to . The cathodic portion of the Butler-Volmer equation contains only the second exponential term in Eq (15.14). For the H+/H2 half-cell reaction it can be written as:

                                      (15.19b)
where
(15.20b)
and                                                                                                               (15.21)

At the intersection in Fig. 15.9, the current densities of the two half-cell reactions are equal. An analytic version of Fig. 15.9 is based on Eq (15.19a) for the metal anodic reaction and Eq (15.19b) for the hydrogen cathodic reaction. Setting  i = icorr and e = ecorr in Eqs (15.19a) and (15.19b), equating their right-hand sides at the intersection and solving for logicorr  yields:

 =                 (15.22a)

solving the last equality for log icorr:
(15.22b)

Fig. 15.9 The corrosion potential and corrosion current from the intersection of cathodic and anodic polarization lines
 
Example Determine the overpotential hcorr and the corrosion current density icorr when M = Ni and: aNi = 0.55, aH = 0.20,  [Ni2+] = 0.02 M, pH = 1, pH2 = 0.1 atm, ioNi = 10-6 A/cm2, ioH = 6´10-9 A/cm2

Nernst potentials from Table 2.2 and Eq (2.77a):
fNi = 0.25 - (0.059/2)log(0.02) = 0.30 V;   fH = 0 - (0.059/2)log([10-1]2/0.1) = 0.19 V
Applying the sign convention:       dH,Ni = -0.19 - (-0.30) = 0.11 V
From Eqs(15.20ab):    
From Eq (15.22b):

Use log(icorr) in Eq (15.22a):   hcorr = 0.054(-5.69 - log(10-6)) = 0.017 Volts
hcorr = 0.11 - 0.037(-5.69 - log(6´10-9)) =0.016  Volts

15.6.4 Tafel diagram for multiple cathodic reactions
In many situations, active corrosion is driven by more than a single cathodic reaction. For example, under neutron or gamma radiation, water radiolysis produces substantial quantities of hydrogen peroxide, which decomposes according to H2O2 + 2e- --> 2OH-.  This half-cell reaction acts in concert with the proton reduction reaction to increase the corrosion rate. The two cathodic lines and the anodic metal dissolution line are shown in Fig. 15.10. The condition requiring equal

Fig. 15.10   Tafel diagram with two cathodic half-cells

rates of production and consumption of electrons are:

ic(H2O2) + ic(H) = ia(M)

To determine the state of corrosion, the dashed line must be moved up from its location without H2O2 (point E), past the corrosion condition if only H2O2 were present (point D) until the combined cathodic lines satisfy the above equation, or:
10OA  + 10OB =10OC
The corrosion current density (point C) is larger than either of the cathodic rates.

15.6.5  Tafel diagram for multiple anodic reactions
A common case of two anodic reactions and one cathodic reaction is zinc plated on iron, which constitutes galvanic protection. Figure 15.11 is an example wherein these two metals exist in the presence of oxygenated neutral water, which provides the cathodic half-cell:                                                  
H2O + O2 + 2e- -->2OH-.
The areas of the two metals exposed to water may be different, so the electron-production/electron consumption balance is:
(AFe + AZn) iO2 = AFeiFe + AZniZn
where AFe and AZn are the areas of the two metals exposed to water. The above charge balance assumes that the cathodic reaction takes place over the entire exposed metal area and that the rates are the same on both metals. With reference to Fig. 15.11, the relevant terms in the above equation are:
iO2 = 10OC        iFe = 10OA                     iZn = 10OB

The corrosion rate of the iron component is AFeiFe  is approaching the exchange-current density, which is ioFe ~ 1 mA/cm2.  As an example, if the corrosion current density at point A in Fig. 15.11 is 10 mA/cm2, the dissolution rate of the metal  can be calculated from Eq (15.16). jcorr can be converted to a loss rate by:
metal loss rate =
which is a significantly smaller loss rate than would afflict unprotected iron.

15.6.6 Concentration polarization
The cathodic polarization curves heretofore represented as straight lines with a negative slope are surface reactions whose rates depend upon delivery of  a dissolved species from the bulk solution to the interface. Like any other reaction at a surface supplied with reactants from a fluid, they are  potentially mass-transfer limited. Consider the cathodic half-cell reaction 2H+ + 2e---> H2.  The transport rate of H+ to the surface depends on the hydrodynamic boundary layer in the water adjacent to the metal and on the diffusivity of H+ in water. Figure 15.12 is a sketch of the H+ concentration close to the surface.

Fig. 15.12  Concentration of H+ adjacent to the metal surface during active polarization

The flux of H+ to the interface and the equivalent current density are:

jMT = kMT([H+]b - [H+]int) = i/F                                                          (15.23)

Since iMT = ic, the cathodic current density, the entire process becomes mass-transfer limited when attains a limiting value iL which reduces [H+]int to zero. The behavior of the cathodic polarization curve with a mass-transfer limitation is shown in Fig. 15.13. The potentials are reported with respect to , the equilibrium interfacial potential difference of the hydrogen half-cell having the bulk hydrogen ion concentration and H2 partial pressure. The overpotential under these conditions is h = 0. The equilibrium interface potential difference for the bulk solution conditions is the Nernst potential :
=  0.059log{[H+]b/ }
where [H+]b is the bulk concentration of hydrogen ions and pH2 is the H2 partial pressure with which the solution is in equilibrium (it need not be 1 atm). Under these conditions, there is no net current but the exchange current density, ioH,  prevails at the interface. When a net current is drawn (e.g., by reducing the overpotential), the H+ concentration at the interface, [H+]int, is no longer the bulk value. The interface potential difference is not:
=  0.059log{[H+]int/ }
The concentration overpotential is defined by:

Eliminating [H+]int by use of Eq (15.23):
(15.24)
Where iL is the limiting current density, meaning that the current density is so large that the entire kinetics is limited by mass transport. This means that [H+]int is reduced to zero.

iL = kMTF[H+]b                                                                                              (15.25)

Fig. 15.13   Cathodic polarization curve with concentration polarization

The exchange-current density also reflects the interface conditions. From Eq (15.19):     
ioH = QH[H+] [pH2]a = QH[H+] [pH2]a                      (15.26)           
QH is the collection of constants in Eq (15.18a) for the hydrogen half-cell reaction. a is also a property of the hydrogen half-cell. However, the H+ concentration and pH2 are the actual conditions of the aqueous phase.

From Eq (15.19b) the overpotential without mass transfer limitation, hc (AB on the dashed line in Fig. 15.13), is ,                                                                         (15.27)        
The overpotential that appears on the ordinate of Fig. 15.13 is the sum:
h = hconc + hc                                                            (15.28)
Example:   show how i and h are related when the concentration overpotential is significant.
[H+]b and pH2 are specified. Let i be the independent variable and h the dependent variable.

1. The mass-transfer coefficient kMT must be known in order to fix iL by Eq (15.25).
2. given i, calculate:
(a) i­oH from Eq (15.26)
(b) hc from Eq (15.27)
(c) hconc from Eq (15.24)
(d) h from (15.28)

15.6.7   Actual Tafel diagrams
Figure 15.14 is a sketch of a realistic Tafel diagram for a metal like iron. Three regions with cathodic polarization curves are depicted. In the bottom of the sketch is the active corrosion region, which has been the sole mechanism dealt with up to this point. Activate corrosion ends at the peak of the anodic curve, for which the coordinates are the critical current density and an overpotential termed the passivation potential. In the middle is a region called passive because the corrosion rate is very low. This is the subject of Sect. 15.7. The region at high overpotential is termed transpassive. It is similar to active corrosion in that the corrosion product is a soluble ionic species. We do not review this type of corrosion because the overpotentials are above that at which water decomposes into O2.

Two cathodic polarization curves are shown in the active and transpassive corrosion zones. For branches  1 and 5, the intersection with the anodic branch of iron lies on exchange-current-density lines. It was shown in Sect. 15.6.1 that io represents equal anodic and cathodic currents and an interface potential difference that is the equilibrium (or Nernst) value. This combination of anodic and cathodic polarization curves is termed thermodynamic control  of the corrosion rate. These polarization-curve intersections are also the corrosion potentials.

It may seem illogical that the corrosion current density should coincide with an exchange-current density, which by definition has zero net current drawn from the equal anodic and cathodic current densities. However, cathodic branches 2 and 4 in Fig. 15.14 are examples of kinetic control of the corrosion process. These are the type that have been discussed at length in the preceding sections. The maximum labeled logicrit on the anodic curve represents a point of
instability.

15.14  Realistic anodic Tafel line with five cathodic line intersections.

If the current density is increased from zero using the cell configuration of Fig. 15.5, the overpotential follows the active corrosion segment of the portion of the curve intersected by cathodic branches 2 and 4. When the critical current density icrit is reached, the overpotential jumps to the transpassive portion of the anodic polarization curve. If the overpotential (rather than the current density) is controlled, again the section from the critical current density to the start of the passification portion of the polarization curve cannot be measured.  If the overpotential is raised slightly above the value corresponding to logicrit, the system again jumps to the transpassive section of the anodic polarization curve.

The region of the anodic polarization curve represented by the vertical straight line in Fig. 15.11 is the region of passivity, so named because of the very low corrosion current density.  In this region the low corrosion rate, for example at the intersection with cathodic branch 3, a protective oxide or hydroxide scale is present on the metal surface. This phenomenon is treated in Sect. 15.7.

Quantitative curves of the anodic polarization curves for iron and stainless steel are shown in Fig. 15.15. Stainless steel is represented by chromium, since this metal is the primary chemical actor for the surface of this alloy in water. Despite the general chemical similarity of Fe and Cr, their anodic polarization curves are vastly different. The primary difference is the critical current densities of the two metals. The very large value of icrit(Fe)  make iron very difficult to passivate. The overpotential corresponding to the critical current density of Cr is an order-of-magnitude lower than that of Fe, which also makes the former easier to passsivate. This difference in corrosion behavior between the two metals is the reason that the iron age morphed into the age of stainless steel and chrome plating.

Fig. 15.15   The anodic polarization curves of iron and chromium (stainless steel)

15.7  Scales on structural metals

15.7.1 Corrosion Properties
By structural metals we mean the iron-nickel-chromium alloys which provide the majority of the high-strength, elastic support of modern society's edifices.  Nearly all of these applications exist at ambient temperatures, with the notable exceptions of fossil-fired and nuclear power plants, where temperatures up to 850oC are involved. The compositions of the three most common alloys are listed in Table 15.2. There is a goodly spread in the
compositions and the table gives rough averages. These alloys have somewhat different corrosion resistance to water and different strength versus temperature dependences. In what follows, only the properties of stainless steel are reviewed.
The result of metal corrosion in water (or air) are oxide scales, which are also called films or barriers.

When a nearly-pure alloy such as Zircaloy corrodes, a single oxide scale is formed.  Stainless steel corrosion in ambient-temperature water, which is very slow, results in a two-layer scale, which is seen in Fig. 15.16.

        Table 15.2  Compositions of Fe, Cr, Ni alloys

            *balance Mo, Ta

Fig 15.16  Duplex corrosion scale on 304 SS exposed to hydrogenated water at 200oC
From S. Ziemniak et al ,Corr. Sci., 44 (2002) 2209 (with permission)
The actual corrosion layer (i.e, that controls the growth rate) is the sub-layer rich in chromite, or in general an oxide whose main metal component is Cr. Lying on top of this corrosion barrier is a collection of particles rich in ferrite. These scales are a class of  mixed double oxides AB2O4 called spinels, where A and B denote mixtures of Fe, Ni and Cr. The sublayer Cr-rich spinel grows into the base metal while the ferrite chunks are deposited from the liquid as recrystallized particles. The reason for this difference is the very low solubility of  Cr in water compared to that of Fe and Ni. This is seen in the compositions of the two layers on various alloys collected in Table 15.3.

  The upper (ferrite) layer contains little chromium, in keeping with the latter's low solubility in water.  The A part of the spinels consists only of Ni and Fe, while the B  component contains no Ni. In these compounds, nickel has a valence of 2+ while Cr is in the 3+ oxidation state. Iron has a valence of 2+ in the A part and in the B portion, the valence is 3+. The range of compositions in corrosion-products spinels is shown below

Table 15.3  Corrosion-films AB2O4, on Fe, Ni, Cr alloys in hydrogenated water at 200oC (after S. Ziemniak et al ,Corr. Sci. 44 (2002) 2209; 45 (2003) 1595; 48 (2006) 2525; 48 (2006) 3330)

Chromite (lower) layer                   Ferrite (upper) layer

 

Recrystallization of a ferrite layer occurs on the zirconium oxide scale growing on reactor fuel-element cladding, with deleterious consequences (see Chap. 21 )                                                                                                                                                                                                                                                              
15.7.2  Parabolic scale growth kinetics
There two mechanisms by which a metal oxidizes. i) metal ions migrate from the metal/scale interface to the scale/solution interface where the ions are converted to an oxide by reaction with H2O or O2 dissolved in water. ii)  oxygen ions produced at the scale-solution interface diffuse through the scale to the metal-scale interface where they extract metal atoms and convert them to metal ions in the scale. Only the second of these possibilities is considered in what follows.
The simplest rate law for growth of a corrosion scale (oxide or hydroxide) on a metal is the parabolic growth law.  This law is based on the flux  of a species of the solid diffusing through the scale according to Fick's law:
(15.29)    
D = diffusion coefficient of species transferring (oxygen ions) in the scale, cm2/s
CL = concentration of O2- at the scale/solution interface, moles/cm3
L = thickness of scale, cm
Co = concentration of O2- at the scale/metal interface, moles/cm3
CM = molar density of the metal, moles/cm3
CM = molar density of  the metal element in the scale, moles/cm3
x = distance from the metal/scale interface (see Fig. 15.2)

The last equality in Eq (15.29) can be integrated to give the scale thickness as a function of time, which is the desired  end result for all of the analyses of corrosion scales. Integrating from L = 0 at t = 0 yields:
(15.29a)                 
which is the parabolic growth law. This law, however, is not consistent with the empirical oxide scaling laws observed on most metals, which are:
L = A + B´ln(t)      or     L = const´t1/n                                            (15.29b)
where A and B are constants in the first form and n is an integer >2 in the second form.
15.7.3 Effect of the ionic character of the metal

 

The above simple analysis does not apply to the oxide scales on metals because of:  i) the ionic character of the species involved in forming oxide scales on metals and ii) the potential difference between the metal and the aqueous solution analyzed in Sect. 15.4. These electrical effects influence the corrosion rate.

In the usual form of Fick's first law, as used in deriving Eq (15.29), the flux is driven only by the chemical potential gradient:
,
where m is the chemical potential of the moving species in the solid. 
According to Eq (2.71),  a position-dependent electric potential provides a gradient which generates a species flux of:

where f is the electric potential, F is the Faraday (96,500 Coulombs/mole or 96.5 kJ/mole-Volt) and z is the charge on the moving species. The total flux (moles/cm2-s) is the sum of the above two contributions:
(15.30)
In an ideal solution, the chemical potential is related to the concentration by Eqs (2.36) and (2.37):  m = go + RTln(C/CM), where go is the free energy of the species in a standard state, The gradient of m is:

The electric potential gradient is expressed in terms of the electric field (Volts/cm):
(15.31)
Replacing dm/dx  in terms of dC/dx and the electric potential gradient in terms of the electric field, Eq (15.30) becomes:
(15.30a)

where                                                                                                                   (15.32)

is the mobility of the moving species (cm2/Volt-s). 

Note  the  fundamental difference between Eq (15.30a) and the Butler-Volmer equation, (15.14). The latter deals with the rates of half-cell reactions, the first term for the anodic (oxidizing) half-cell and the second term for the cathodic (reducing) step. These processes are driven by differences between an applied potential and the Nernst equilibrium potential. Equation (15.30a), on the other hand, describes the flux of a species through an oxide film. The movement is driven by spatial gradients of the potentials.

15.7.4  An Illustrative System - the oxide scale on a divalent metal
The reason for choosing a divalent metal, which could be (among many others) Fe or Ni, is to keep the notation simple. The oxide is written as MO and is stable on the metal (i.e., does not dissolve in the aqueous phase) because the metal concentration in the solution is at or above the solubility limit (Sect. 2.9.5):
MO(s) + H2O =  M2+(sol'n) +2OH-

Allowing for deviations from exact stoichiometry, the oxide has the formula MO1-y. For simplicity, only hypostoichiometry is allowed. The nonstoichiometry parameter y is equal to the fraction of anion lattice sites that are vacant. If anions are missing but all cation sites are filled (as assumed here), electrical neutrality is maintained by leaving behind the two electrons formerly associated with the missing oxygen ions. Where do these electrons go? As shown in Fig. 15.17, they move to the oxide/solution surface where they reduce neutral oxygen in the solution to anions (O2-).

In nearly all cases, oxygen ions migrate by the vacancy mechanism, which means that the flux of oxygen ions through the oxide can equally well be considered to be movement of anion vacancies in the opposite direction. Vacancies cannot be the sole species migrating from the oxide-metal interface to the solution-oxide interface, for this would be equivalent to  removal of positive charges from the metal. The resulting buildup of the negative charges in the metal would eventually stop the movement of O2- and corrosion would cease. This charge flux is counteracted by the equal and opposite flux of diffusing electrons.

This illustrative corrosion mechanism can be described by the following electronic processes:

at the metal/oxide-scale interface (x = 0):              M(met) --> M  + 2e' (met/ox)
in the oxide film (or scale):                                   e- (met/ox) --> e- (ox/sol'n)
and

at the oxide-scale/sol'n interface (x = L): 

         
Kroger-Vink notation
Identification of the defects in the above reactions follows Kroger-Vink notation. In this method, the letter signifies the type of defect: V for vacancy, I for interstitial. The subscript indicates the location of the entity: O means on the anion  (oxygen) sublattice, M means the cation (metal) sublattice, etc. The superscript indicates the charge deviation from the perfect lattice, dots .  for positive charge and apostrophe ' for negative charge. Accordingly, the symbol  in the above reactions and in Fig. 15.17 identifies a vacancy (V) on the anion sublattice (sub O) which is doubly-positively-charged relative to the same site occupied by O2- (superscript..). e' symbolizes an electron with a single negative charge. A missing charge symbol indicates species on normal lattice positions (MM means a cation on the cation sublattice and OO denotes an anion on the anion sublattice). The terms in parentheses give the macroscopic location of the defect.

Whatever the mechanism , the overall corrosion reaction in this example is the sum of the three steps:
M(met) + O2(sol'n) = MO (ox)

A key step that requires elucidation is the movement of O2- from the scale/sol'n interface to the metal/scale interface so that it can occupy a lattice position adjacent to an M2+ ion. 

If  oxygen diffuses by a vacancy mechanism (Chapter 5, section 5.4) on the anion sublattice, the anion diffusion coefficient is DO = DVOcVO, where DVO is the diffusivity of the anion vacancies and cVO is the vacancy fraction in the anion sublattice. The sum of the vacancy fraction and the O2- fraction equals one. Hence a gradient in the O2- concentration generates an opposing gradient of the anion vacancies. If cVO varies through the scale, the  oxygen diffusivity DO is position-dependent as well. This would complicate the analysis. However, the vacancy diffusivity DVO is constant throughout the scale, so in the present analysis, the anion vacancy is chosen as the moving species instead of oxygen ions. However, anion vacancy diffusion from the met/ox interface to the ox/sol'n interface is equivalent to anion diffusion in the opposite direction, so the choice of the former is merely a matter of convenience.

The remaining question is how the anion vacancies are produced at the metal/oxide interface. Formation of this point defect at first seems counterintuitive, given that O2- ions are moving towards this interface. The explanation is twofold:  i) the O2- anions move by hopping into existing vacant anion sites, which are those moving in the opposite direction, as explained in the previous paragraph; ii) the very act of creating M2+ cations in the lattice structure automatically

Fig 15.17  Schematic of a metal with an oxide scale immersed in an aqueous solution

creates vacant anion sites in order to preserve the crystal structure. This step is illustrated in           Fig. 15.18, where again for the purpose of illustration, the MO crystal is assumed to be the NaCl-type (Fig. 2.9).

Figure 15.17 includes a sketch of the variation of the electric potential, f, in the three phases. Note the discontinuities at the two interfaces, which are due to unequal charges on either side of the interfaces. Also, the variation of the potential in the scale is not a straight line, as assumed for the double layer in the absence of the intervening oxide (Fig. 15.2).  As we shall see, this curvature causes severe complications in the analysis of the corrosion process.

Electrons can move through an oxide scale by the quantum-mechanical process called tunneling, whereby an electron "passes through" an energy barrier that it could not surmount by the usual thermal agitation. At higher temperaturers and thicker scales, the electrons move by hopping from one energy minimum to an adjacent one. This process is essentially diffusion, and is described by the same equations as diffusion of atomic species (i.e., Eq (15.30a).

Fig. 15.18  Adding cations to an oxide with the NaCl structure

The objective of the following analyses is to calculate the flux of species (oxygen vacancies in this case) that is responsible for growth of the scale. The rate of film growth is determined from:

                                                                 (15.33)

where CM is the molar density of the cations in MO, which is equal to the sum of the anion vacancy and filled anion-site concentrations. The subscript VO means vacancy on the oxygen sublattice, or anion vacancy.  CM is the density of the metal component (cation) of the oxide.

Once JVO is known as a function of L, Eq (15.33) can be integrated to give the growth law,         L = f(t). 

15.7.5 Position-independent electric field
For the electric field to be constant throughout the scale, Eq (15.31) requires that the electric potential vary linearly with position in the scale. At steady-state, the flux of anion vacancies through the scale (JVO) is a constant, and  provided that the electric field E does not depend on x,  Eq (15.30a) can be integrated with the boundary condition CVO = CVO(0) at x = 0 (the scale/metal interface):
(15.34a)

where                                                    (15.34b)

zVO = +2 is the charge on anion vacancies (relative to the perfect lattice) F = 96.5 kJ/mole-Volt is Faraday's constant. With the boundary condition CVO = CVO(L) at x = L (the scale/sol'n interface) in Eq (15.34a), solving for JVO yields:
(15.35)

In the limit as a-->0 (or E-->0), Eq (15.35) reduces to Eq (15.29).

Using Eq (15.33) the growth rate of the scale is:

                                                           (15.36)

Time-integration of Eq (15.36) hinges on how the electric field varies with L. Since E is constant across the scale, the electric condition of solid scale is analogous to that across the double layer as depicted in Fig. 15.2. This condition is described by:
(15.37)
If  the electric potential difference (the numerator of the above equation) is independent of scale thickness, E varies as L-1.  With a given by Eq (15.33b), the argument of the exponential functions in Eq (15.36) is not a function of L. However, the coefficient varies as L-1.  Integrating Eq (15.36) with L = 0 at t = 0 yields the parabolic growth law but with a different coefficient from that of Eq (15.29a):

                        (15.38)

As Df --> 0, Eq (15.38) reverts to Eq (15.29a). Problem 15.2 compares rate constants for the two cases.

15.7.6  Nonparabolic scaling kinetics - constant electric field
If instead of holding Df constant as the scale grows, the electric field E retains a fixed value, integration of  Eq (15.36) gives:
(15.39)

where  x = CVO(L) /CVO(0).   Even for small values of L, the exponential term in this equation is much larger than x, so the Eq (15.39) reduces to:
(15.39a)
According to this equation, L should increase linearly with time. This is contrary to experimental observations of corrosion kinetics, so the constant-field model is rejected.

15.7.7  The effect of space charge in the oxide scale
If  the electric potential does not vary linearly through the scale, integration of Eq (15.30a) becomes considerably more complex.

To avoid continual charge buildup or depletion in portions of the scale, the net charge crossing any plane parallel to the  interfaces must be zero, or at all x:
Je = 2JVO                                                        (15.40)
In the notation of Eq (15.34b), the flux of anion vacancies is given by Eq (15.30a) as:
(15.41)

The electric current flowing through the scale consists of components due to the diffusing O2- (or, what is equivalent, the oppositely-directed anion vacancies as expressed by Eq (15.41)) and that of the electrons (subscript e). The latter is:
(15.42)
Although the net charge flux is zero (Eq (15.40)), sections of the scale perpendicular to the interfaces may build up a net electrical charge. This is termed the space charge­, and is related to the electric field by Poisson's equation:
(15.43)

Here Faraday's constant is F' = 96,500 Coulomb/mole. e is the permittivity of the oxide, ~10-12  Coulomb/Volt-cm. The term in parentheses multiplied by F' is the space charge (units of Coulombs/cm3). 
The three coupled first-order ordinary differential equations, Eqs(15.41), (15.42) and (15.43) require initial conditions (at x = 0, the metal/scale interface). To this end, it is assumed that structural equilibrium (i.e., the M/O ratio) between adjacent phases prevails at both interfaces.

In terms of the deviation from exact stoichiometry MO1-y, the relation between the O/M ratio 1-y and the anion vacancy concentration is CVO = yCM. The stoichiometry deviations at the two interfaces can be obtained from the M-O phase diagram, as shown pictorially in Fig. 15.19. The intersection of the isotherm with the lower phase boundary of the MO1-y phase (point A) is the y-value at the metal/oxide interface. The intersection of the isotherm with the isobar representing the system's O2 pressure (point B) is the corresponding value at the oxide/solution interface. These give CVO (0) = yACM and  CVO(L) = yBCM, respectively, for the bounding anion-vacancy concentrations for Eq (15.41). 

Ce(x) and E(x) need to be provided with values at x = 0. What then remains is a solution method, which is inevitably numerical. Below we simplify the system as much as possible and provide such a solution.

 15.7.8  Solution of the equations for oxide-scale growth with space charge
The relation of the various quantities derived in the previous section are best seen if cast in dimensionless form. The dependent variables are:

                         (15.44)

The constants in the above definitions are:

 The dimensionless forms of the anion-vacancy flux through the oxide scale and its thickness are:
(15.46)
In terms of the above dimensionless variables, Eqs (15.41) - (15.43) are:

                                                               (15.41a)
(15.42a)

                                                             (15.43a)

Fig. 15.19  Hypothetical phase diagram for the M/MO system.                                                          The dashed line is an O2 isobar; the red line is an isotherm

The factor of 4 in Eq (15.42a) is twice the ratio DVO/De, with the diffusivities estimated by Fromhold7.

The boundary conditions for Eq (15.41a) are:

                              (15.47)

where yA and yB are the nonstoichiometry parameters characterizing the oxide at its two extremities  (Fig. 15.19). Generic concentrations at these locations used by Fromhold are:

yA = 4´10-6           yB = 4´10-7                                              (15.48)

When inserted into the first equality of Eqs (15.44) the bounding conditions on the anion vacancy concentrations are:

qVO(0) = (36)(1.1´1017)(0.1)(4´10-6)(10-14) = 0.015L2        and      qVO(1) = 0.0015L2      (15.49)

The unit of the oxide scale thickness L is nanometers. The factor of 10-14 converts nm2 to cm2 
Initial conditions for Eqs (15.42a) and (15.43a) are also required. For the former, Fromhold7 recommends Ce(0)/CM = 4´10-5, although this is not an experimentally-determined value. Consequently, this ratio needs also to be treated as an undetermined condition. 
Finally, the initial condition for Eq (15.43a) is given by the last equality of Eq (15.44) with                   E = Eo. This electric field depends on the separation of charges between the metal and the oxide
at their common interface. There is no a priori method of assessing this quantity, although many
sources argue that Eo ~ 105 - 106 Volts/cm. Once a value of this parameter is specified, the initial condition for Eq (15.43a) is
(15.50)
Solution of the three coupled ordinary differential equations is accomplished by a trial-and-error method:  for a fixed L, JVO (in its dimensionless form j from Eq (15.46)) is guessed and  integration is performed numerically starting from h = 0. When h = 1 is reached, the calculated qVO(1) is compared to the value given in Eq (15.49). When the two agree, the correct value of JVO has been chosen.

However, the presence of the poorly-known parameters qe(0) and Eo greatly complicates this procedure. In actuality, all three unknowns, the above two quantities and JVO, must be simultaneously guessed. After integration, the error measure:
error =   |qVO(1)(calc) - qVO(1)|/q VO(1)                    
is computed. When this is less than 5%, the trio of parameters forms an acceptable solution to the three equations.

As expected, at a fixed value of the scale thickness L, this procedure results in a large number of acceptable  parameter combinations. However, these all fall within finite ranges, and when plotted as distributions, provide average values that appear reasonable. For example, Fig. 15.19 shows the results of the parameter search for L = 5 nm for the anion vacancy flux JVO (or j of Eq (15.46)). From Fig. 15.19, the weighted average value of the dimensionless flux,

is 0.082 ± 0.4. While not particularly accurate, the 50% error is the best that can be obtained for a single specified parameter (qVO(1)) fitted by selection of three parameters.

When the  values for other scale thicknesses are computed in the same manner and converted to JVO values using Eq (15.46), the JVO vs L plot shown in Fig. 15.20 results. In converting  to JVO, the anion vacancy diffusivity was taken to be DVO = 4´10-11 cm2/s at 300 K.

The best-fitting line through the points is:
(15.51)
where A = 7.5´10-11 moles/cm2-s and  b = 0.47 nm-1.  L is in nm.
The thickness dependences of the other two fitted parameters are shown in Table 15.4

Table 15.4  Fitted parameters in the oxide at the metal/oxide interface; electron concentration (2nd column) and the electric field (3rd column)

The fitted values of the electron concentration at x = 0 (2nd column) is very close to Fromhold's guess7. In addition, it is nearly independent of scale thickness. The metal/oxide interface electric field is below the commonly-reported value of 105-106 Volt/cm and is also essentially independent of scale thickness.

Fig. 15.19  Distribution of acceptable anion-vacancy fluxes for a scale thickness of 5 nm

Lastly, Fig. 15.21 shows the spatial distributions at L  = 3.3 nm of the three properties that characterize the oxide scale. The anion-vacancy concentration (CVO/CM) decreases from the oxide/metal interface (x/L = 0) to the oxide/solution interface (x/L = 1). This behavior is forced on the solution as the conditions at these two locations are prescribed.

Fig 15.20  Dependence of the anion-vacancy flux on scale thickness

The initial values of the electron concentration and the electric field, on the other hand, are chosen as part of the random parameter-search process. The electron concentration at x  = 0 is very close to the value suggested by Fromhold7. The x-variation of this concentration, however, requires some explanation. In order for electrons to move from the metal/oxide interface to the oxide/solution interface, Eq (15.42a) suggests that the concentration gradient should be the opposite of that shown in Fig. 15.21; that is, the electrons move up their concentration gradient instead of down it. The reason for this behavior is the strong coupling between Eqs (15.42a) and (15.43a); qe starts out about an order-of-magnitude larger than qVO, so that the latter can be neglected in Eq (15.43a). Thus the gradient of the dimensionless field Y is always negative. As a result, the second term on the right-hand-side of Eq (15.42a) quickly dominates the 4j term and the gradient of qe becomes positive. As qe grows ever more positive, Y becomes increasingly negative, which, according to Eq (15.42a), accelerates the rise of qe with distance into the scale. This feedback phenomenon is clearly evident in the two dashed curves in Fig. 15.21 and is observed for all scale thicknesses.

15.7.9  Scale growth rate
Having determined the dependence of the anion-vacancy flux on scale thickness (Eq (15.51)), we are now able to integrate Eq (15.33):
L(nm) = 2.1´ln(3.6´10-3t + 1)                                  (15.52)

t is in seconds. This equation is plotted in Fig. 15.22. For the anion-vacancy diffusivity assumed for this calculation (DVO = 4´10-11 cm2/s), a scale thickness of 10 nm is achieved in ~ 10 hours.
For comparison, a parabolic scale-growth curve from Eq (15.29a) is plotted in Fig. 15.22. The sharper curvature of the solid curve more closely resembles experimental data for scales in the size range of the graph and formulae such as Eq (15.29b). Figure 15.22 shows that the magnitude

Fig. 15.21  Variation of the concentrations of the anion vacancies and electrons and of the electric field with position in the scale.  L = 3.3 nm

of the parabolic line is everywhere much greater than that from Eq (15.52). The reason for the inhibiting effect of the electric field on the corrosion rate lies in the second term of Eq (15.30a). The electric field curve in Fig. 15.21 is negative over 90% of the scale thickness, which means that the second term in Eq (15.30a) diminishes rather than increases the flux of anion vacancies from the metal/oxide interface to the oxide/sol'n interface. This is equivalent to reducing the flux of oxygen ions in the opposite direction, and hence decreases the rate of formation of metal oxide.

15.8   Passivity

A scale forms on a metal when solution conditions (i.e., pH and overpotential) are in the passivity  region of the polarization curve of Fig. 15.14. The film is called "passive" because it persists indefinitely and protects the metal from rapid corrosion. The phenomenon is extremely important for structural metals, all of which, in air or water, should (thermodynamically) be converted back to the ore whence they originated. In particular, chemical immunity of aluminum, chromium and stainless steel from attack by water, even at room temperature, is due to the formation of passive films. Because these scales are no more than a few nanometers thick, they can only be seen with appropriate microscopes.
Instead of ascribing the onset of formation of an oxide scale to the metal ion in solution reaching its solubility limit (Sect. 15.7.4), connection to Tafel diagrams is made by considering the appropriate half-cell reaction. For the MO solid oxide treated in Sect. 15.7, this is :

M + H2O = MO + 2H+ + 2e-                                                 (15.52)

The maxima in the polarization curves in Fig. 15.15 (at logicrit) are reached at the passivation overpotential . For the half-cell reaction (15.52), this is:
hpass =  + 0.059log[H+]2 = - 0.118pH                                 (15.53)

 is the passivation overpotential at pH = 0. Note that for the pH, the H+ concentration is in units of molarity, or moles per 103 cm3 (a liter). Thus, in a solution with pH =7, the H+ concentration is 10-10 moles/cm3.

Description of passivation by a Butler-Volmer equation (also called the Tafel equation) such as Eq (15.14) applies to an electrochemical cell in a laboratory in which the overpotential given by Eq (15.53) is applied and the current measured (or vice versa). However, the application with which we are concerned is an aqueous phase in an industrial device, where the electric potential is determined by many factors that cannot be controlled. In this case, scale growth (corrosion) and  passivation are treated  as kinetic phenomena, which are best described by the methods in Sect. 15.7.

The mechanism is straightforward: scale growth stops at a thickness where the rate of the scale-growth reaction at the metal/oxide interface equal the rate at which the oxide dissolves at the oxide/sol'n interface. The former is given by the combination of Eqs (15.43) and (15.51):

Dissolution is assumed to proceed at the oxide/scale interface by the reaction:

MO + 2H+ --> M2+(sol'n) + H2O                                           (15.55)
at a rate given by:
rate = krxn[H+]2,  moles/cm2-s                                                (15.56)

where krxn is the rate constant for reaction (15.54).  In this equation, the concentration must be expressed in units of moles/cm3.

 The rate of shrinkage by dissolution of the scale is:

The scale achieves a steady-state thickness, LSS given by the condition that the net rate of change is zero:

from which LSS is found to be:

                                              (15.58)
Example:  a piece of the metal is immersed in a solution of pH = 10
The rate constant of reaction (15.54) for a particular metal is krxn = 2´109

(a) what are the units of krxn?
(b) what is the thickness of a passive scale on the metal?
(c) at what rate does the scale penetrate the underlying metal?

(a)  With the H+ concentration in units of moles/cm3, Eq (15.56) gives:

(b) From Eq (15.58):
= 3.0 nm
(c) from Eq (15.54)

 or, the metal is corroded at a rate of only ~ 0.006 nm/yr! The passive scale on this particular metal is indeed very protective.

The corrosion current density is associated with the flux of oxygen ions through the scale to the metal/oxide interface, which is where the conversion of metal to metal oxide occurs. With the oxygen ion flux given by Eq (15.51), the corrosion current density is also equal go the rate at which electrons arrive at the oxide/solution interface:

icorr = 2F'JVO = 2F'Aexp(-bLss)   A/cm2                                 (15.59)

where F' = 96,500 Coulombs/mole is Faraday's constant. For the 3-nm-thick passive scale of the preceding example, the corrosion current is 3.5 mA/cm2.

15.9  Localized corrosion

Until now, only uniform corrosion has been treated. If this were the only form of the electrochemical attack of metals, our industrial society would be relieved of an significant portion of its problems. Not all uniform scales are as protective as passive films. One exception are oxide scales that reach a thickness where they crack because of stresses arising from the difference in the metal-component density between the metal and the oxide. Cracking allows the solution ingress to the bare metal, and corrosion resumes. Zirconium alloys (e.g., Zircaloy) exposed to 300oC exceptionally-clean water in water-cooled nuclear reactors suffer from this problem.

All structural metals (iron, nickel, chromium, aluminum, titanium, zirconium, etc) are chemically reactive and when exposed to water or air, would much prefer to obey their thermodynamic tendency and convert back to the oxides whence they came. It is only a matter of time that they do so. The challenge is to find ways of deferring this conversion as long as possible. Several means of so doing include:
1. keep the temperature as low as possible. Sometimes high temperatures cannot be avoided. (e.g., piping in electric power plants)
2. develop alloys of the metals that are more resistant to corrosion than the elemental metal. Adding chromium to iron produces stainless steel, so named because of this property. Alloying additions improve the corrosion resistance of pure zirconium metal (see Chap. 17).
3. avoid aggressive anions (e.g., Cl-) in the water.
4. Galvanic protection

A particularly uncompromising form of attack is called localized corrosion. As the name implies dissolution of the metal occurs in isolated spots. Also, the direction of the process is into the, metal, rather than on its surface which is the result of uniform corrosion. There are two distinct forms of localized corrosion, pitting and crevice corrosion.

15.9.1  Pitting
Pitting is initiated by penetration of  the passive film on the surface of the metal by one of three mechanisms:
1.  movement of an aggressive anion (esp. Cl-) through the passive scale by diffusion, without leaving macroscopic signs. This process is facilitated by high electric potential differences across the scale and/or a scale which is amorphous rather than crystalline
2.  breakage of the passive scale by any one of a number of causes, including metal nonuniformities, grain boundaries, surface roughness which are mimicked by the scale. Weak points in the passive scale can progress to local rupture by temperature or stress changes of the component. Penetration of reactive anions initiate pit growth
3. adsorption of aggressive anions on the surface of the passive film. Their negative charge pulls metal ions from the inside of the film. In addition this mechanism tends to thin the scale locally, eventually forming an embryo pit.

There is no established theory that would enable analysis of pit initiation, nor is there any for pit growth. The reasons are twofold: first, the geometry of  pits is so variable that no single shape fits; second, the diameter-to-depth ratio is near unity, which means that convection due to moving water influences their growth.These difficulties are illustrated by the large range of pits observed on metals that are shown in Fig.15.23.

Perhaps the most astounding corrosion pit was found in the pressure-vessel head (top lid) of the Davis-Besse nuclear power plant in 2002.  A drawing of the head and a sketch of the hole are shown in Fig. 15.24. A leak of the reactor coolant (high-purity water with boric acid) through fittings in the control-rod drives initiated the pit. A crust of solid boric acid on the outer surface of the head hid the corrosive attack from inspection. Over a period of several years, the concentrated boric acid solution inside the hole generated the pit in the carbon-steel main body of the pressure-vessel head. Corrosion ceased when the pit reached the stainless-steel liner on the inner surface of the head. The purpose of this liner is to prevent pitting corrosion of the head from the high-pressure (15 MPa), high-temperature (300oC) coolant water with as much as 0.2 M dissolved H3BO3 filling the  inside of the vessel. This gigantic pit is shown in Fig. 15.25.

Fig. 15.24(a)  From Davis-Besse Reactor Vessel Head Degradation            Lessons-Learned Task Force Report, September 20, 2002

 

In view of the enormous range of sizes and shapes and the complexity of the electrochemistry involved, it is no surprise that no quantitative modeling of the growth kinetics of pit corrosion exists. However, detailed qualitative accounts of pitting corrosion can be found in Ref. 2.

 

 

Fig. 15.24(b)  Corrosion pit. from: Davis-Besse Reactor                                        Vessel Head Degradation Lessons-Learned Task Force                                   Report, September 20, 2002

 

Fig. 15.25.   Corrosion hole (pit) in the pressure-vessel head of the Davis-Besse reactor.

 

15.9.2  Crevice corrosion
Crevice corrosion, a close cousin of pitting corrosion, occurs in geometrically simple confines, and so is a favorite target of corrosion modelers. This form of corrosion occurs on close-fitting metal parts that are purposely or accidentally exposed to water containing aggressive anions. Typical venues for crevice corrosion include bolts, valves, tube holders (as in steam generators) and welded joints. The flat sides of a crevice are usually the same metal. However, there are crevices formed from two dissimilar metals or with on side a metal and the other a nonmetal. Examples of the latter are a gasket pressed against a flange or a metal piece pressed against a metal coated with an oxide. 

As opposed to the variety of irregular shapes of pits, the geometry of crevices is regular and well-suited to analytical modeling. Crevices have a well-defined depth (or length L), a constant width (w), or other simple shape and a straight third dimension with no property or condition variations. The crevice opens to the bulk water at its mouth.
A generic crevice is shown in Fig. 15.26.

Fig. 15.26  Crevice in a metal under water

The sequence of steps that drives the corrosion processes inside a water-filled crevice include:

1. O2 in the water in the crevice is depleted by reaction with the metal. The close proximity of the metal sides means that replenishment of O2 requires molecular diffusion from the crevice mouth through the stagnant water filling the length of the crevice, which is slow compared to the corrosion reactions.

2. Due to the restricted geometry, dissolution of the passive oxide film by reaction (15.55) on the sides and bottom of the crevice builds up the concentration of Mz+ in the crevice. Because of the slow transport by diffusion to the crevice mouth the metal ions cannot easily escape to the bulk water.

3. The high concentration of metal ions drives hydrolysis reactions, such as                                              M2+ + H2O --> M(OH)+ + H+ eventually leading to precipitation of a solid hydroxide or oxide when the ion reaches its solubility limit, e.g, M(OH)+ + H2O --> M(OH)2 + H+. The net result of these reactions is twofold: i) the solid that precipitates inside the crevice increases the resistance to transport to and from the crevice mouth. ii) H+ ions are produced with consequent decrease in the pH of the solution at the base of the crevice. The period between the time of O2 depletion and generation of a low-pH electrolyte in the crevice by this mechanism is termed incubation.

4. The net result of metal dissolution in the crevice is generation of an excess of positive ions (M2+ and H+). Unless these are electrically neutralized by anions, dissolution ceases. This is where the "aggressive" anions in the bulk solution come in; if the bulk solution contains a significant concentration of NaCl, for example, Cl- is available to diffuse down the crevice to neutralize the excess of cations.  This permits the corrosion reaction to continue.

5. The high acidity in the crevice moves the solution in the crevice to the low-pH end of the Pourbaix diagram (Fig. 15.1), where the stable corrosion product is M2+, not an oxide or hydroxide that would passivate the metal. This can be also be shown for the half-cell reaction
for which Nernst potential is given by Eq (2.77a). According to the note adjacent to Fig 15.14, decreasing the pH moves the cathodic line so that it crosses the metal's anodic polarization curve in the active corrosion zone. As a result, the corrosion mechanism changes from passive to active, with the concomitant increase in the rate of attack of the metal.

6. The electrons produced by the anodic half-cell reactions on the crevice walls or at the bottom of the crevice are conducted through the metal to a surface contacted by the bulk water. Here O2 is readily available to the metal from the bulk water, so the electrons are consumed by the usual cathodic half-cell reaction: O2 + H2O + 2e- --> 2OH-.

For the following analysis , the crevice processes are simplified as shown in Fig. 15.26.
1. The width of the crevice is constant

2. The external surface of the metal and the walls of the crevice corrode in the passive mode, but this source of metal ions in solution is neglected here.

3. Only the bottom of the crevice is actively corroding - this is the locale of the anodic reaction
M --> M2+ + 2e-
4. The electrons generated at the crevice bottom are conducted to the external surface of the metal where there is sufficient O2 in the water  to permit removal of the electrons by the cathodic reaction.

5. The electric potential (charge) of the metal is fM and that of the bulk solution is fbulk.
The electrode potential is the difference between the charge on the metal and the charge in the solution. When measured by a standard hydrogen electrode (SHE), as shown in Fig. 15.8, the corrosion potential of the bulk water is: ecorr = fM - fbulk

6. The corrosion potential is constant throughout the bulk water (outside of the crevice); specifically it persists over the crevice mouth

7. All processes are in steady-state

8. The problem contains seven unknowns, six concentrations:
a). M2+  b.)  M(OH)+   c.) Cl-  d.) Na+   e) H+   f.) OH-
and the electric potential of the solution in the crevice, fsoln. These variables are functions of depth in the crevice (x) only

9.  Precipitation of solid corrosion product by the reaction: M(OH)+ + H2O --> M(OH)2 + H+ is neglected.

Mass transport of ions in the crevice solution (letter designation in No. 8 above)
The fluxes of the six species in the crevice solution are generated by both concentration gradients and an electric potential gradient. The transport equation for ions in the liquid is the same as that in solids (Eq (15.30). With the electric field replaced by the gradient of the potential in                         Eq (15.31), the flux of species i is:
,
and ui is the mobility of species i:
(15.32)
F is Faraday's constant in units of J/mole-Volt.  Ci is the molar concentration of i (used in place of [i] for convenience)

 

Relations between fluxes:
metal (as M2+ or M(OH)+):                            jM = ja + jb                                                          (M)

jM is the flux of ions containing M2+, which, as shown below, is obtained from the Tafel equation for active corrosion at the crevice tip.   

Water: all hydroxyl (as OH- or M(OH)+) and H+ originate as H2O: 
jb + jf = je                                                                                 (HOH)
Relations between electric potentials
In the bulk water outside the crevice (and at the crevice tip x = 0), the electrode potential is:
ecorr = fM - fbulk                                                         (15.33a)
ecorr is measured by a cell such as the one shown in Fig. 15.8. In the crevice, the electrode potential is:
esoln = fM - fsoln = ecorr + (fbulk           - fsoln)                                     (15.33b)

In principle, esoln could be measured by connecting the liquid in the crevice at depth x to an SHE via a liquid bridge, but of course this is not possible.

Electrochemistry in the crevice
Substituting Eq (15.60) and Eq (15.33b) into the metal flux relation, Eq (M), gives:                               
(15.61)

In our simplified crevice-corrosion model, active corrosion takes place only at the crevice tip. Here, the Tafel equation for the metal corrosion current generated by the half-cell reaction                  M --> M2+ + 2e-  is obtained from Eq (15.19a) with the overpotential given by esoln as defined above:  

where esoln(L) is the electrode potential at the crevice tip (x = L).   jM is negative because dissolution of the metal creates a flux directed away from the crevice tip, or in the negative  x direction. The exchange current density ioM is converted to a metal flux by the factor 2F', where F' is Faraday's constant in units of Coulombs/mole. The factor of 2 accounts for the valence of the metal in this example.     jM is independent of x, which means that the metal-containing ions are leaking out of the crevice into the bulk solution.

The second equation of the required seven is obtained by substituting Eq (15.60) into Eq (HOH) and replacing fsoln with esoln using Eq (15.33b) above. The result is:

                                   (15.63)

For the Na+ and Cl- ions, the flux equations are:
(15.64a)
and
(15.64b)
Note: the signs before the mobilities reflect their valences (+ or -).
Because neither of these ions takes part in an electrochemical reaction with the metal at the crevice tip, the right hand sides of these two equations are zero.

Additional flux equations are not necessary because chemical equilibrium relates concentrations. The first reaction is:
M2+ + H2O = M(OH)+ + H+                                                 
for which the law of mass action is:
KM  = CbCe/Ca                                               (15.65M)

The second is the usual equilibrium between H+ and OH-:

H2O  = H+ + OH-                                                      
for which:
KW = CeCf                                                     (15.65W)

The final equation is a consequence of the requirement of electrical neutrality:

2Ca + Cb - Cc + Cd +Ce - Cf  = 0                                             (15.66)

The boundary conditions at the crevice mouth require the concentrations to reflect the composition of the bulk water:
Ca(0) = Cb(0) = 0                                           (15.67a)

Cc(0) = C4(0) = CNaCl                                     (15.67b)

Ce(0) = 10-pH                                                  (15.67c)

esoln(0) = ecorr                                                 (15.67d)

Input information
i) The salt concentration CNaCl and the pH of the bulk solution.
ii) Equation (15.65W) provides Cf(0).
iii)The corrosion potential of the bulk water, ecorr.
iv)The diffusivities of the ionic species in water are reasonably well known, as are                                    v) The equilibrium constants KM and KW.
vi) The shape of the crevice (width w and length L)
vii) metal composition and electrochemical properties

 

Solution method
The system of 7 equations is highly nonlinear and must be solved numerically. In addition,                           trial-and-error is necessary. Specifically, the electrode potential at the bottom of the crevice, esoln(L) is not known a  priori  and must be guessed before the numerical solution can be applied. Sequential guessing of this quantity continues until the guesses converge.

 

Computational results
Figures 15.27 and 15.28 present typical results of the above analysis of the pertinent features of the crevice. Figure 15.27 shows all ion concentrations as a function of distance from the crevice tip (direction opposite to that of x used above). The OH- concentration is obtained from the H+ concentration with the aid of Eq 15.65M. The calculations behind the curves in this plot included active corrosion along crevice walls, which was neglected in the model presented in this section. The important features of the curves in Fig. 15.27 are:
(a) sharp changes occur within 0.1 mm of the crevice mouth; by 0.2 mm the curves become essentially flat, indicating no further concentration changes.
(b) The pH is ~ 4.5 over most of the length of the crevice.
(c) The combination of the increases in the concentrations of Fe2+ and Fe(OH)+ along the crevice causes a high Cl- concentration to provide solution neutrality. Na+ has been pushed out of the crevice.

Fig. 15.27  Concentration profiles in a 2 mm long crevice with the following conditions: emouth = 0 Volts; CNaCl = 0.02 M; pH = 7. Active corrosion on walls permitted.                           From Sharland (footnote 11)

The profiles of the electric potential in the cavity in Fig. 15.28 show a decrease of                     ~ 0.3 Volts moving from the crevice mouth to the tip, with the drop larger for active walls because of the faster corrosion rate than for passive walls. The consequence of the simultaneous decreases in pH and potential can be seen on the Pourbaix diagram. If the initial condition of the water in the crevice were pH = 7 and a potential of +0.2 V, the point with these coordinates in Fig. 15.1 would lie in the Fe2O3 region. In the crack, the pH drops to 4.3 and the potential to            -0.1 V, which moves the point into the Fe2+ region. This is what permits active corrosion on both the crevice tip and the walls.

Fig. 15.27   Potential rofiles in a 2 mm long crevice with the following conditions:               emouth = 0 Volts; CNaCl = 0.02 M; pH = 7. Active corrosion and passive walls.                           From Sharland (footnote 11)

Improvements of the model to account for:

i) a passive-corrosion flux from the crevice walls that supplies M2+  to the liquid (Eq (15.55));
ii) cathodic half-cell reactions (e.g., H+ + e- --> H2)  along the crevice walls
iii) breakdown of passive corrosion on the walls which then undergo active corrosion                 (e.g. M --> M2+ + 2e-);
iv) aggressive anions other than Cl- in the bulk water (e.g., );                    
v)  impeding diffusion of ions in the water in the crevice by precipitation of metal hydroxides         (e.g. by the reaction M(OH)+ + H2O --> M(OH)2 + H+) - the solubilities in 25oC water of ionic species of iron, nickel and chromium are very low;
vi) concentration and electric potential gradients in the bulk water, meaning that Eq (15.67d) is no longer valid.
vii) trapezoidal rather than the rectangular crevice shape shown in Fig. 15.26. This modification is necessary when the method is applied to a crack in the metal rather to a crevice formed by a gap between two metal pieces.
viii) transient crevice corrosion
ix)  additional homogeneous reaction than (15.65M) and (15.65W) (e.g., M2+ + 2Cl- = MCl2)

References

1. J. Newman, "Electrochemical Systems", pp. 12 - 21, Chaps. 3, 8, 11 and 16, Prentice-Hall Inc. (1973) and later editions.

2.  P. Marcus, Ed, Corrosion Mechanisms in Theory & Practice", 2nd Ed, Chaps. 1, 6, 8, 11, m 12,  Marcel Dekker (2002)

3.  J. Scully, "The Fundamentals of Corrosion", 2nd Ed. , Chap. 2, Pergamon Press (1981)

4. P. Atkins, "Physical Chemistry", pp. 326 - 331, Chap. 29, W. H. Freeman (1978)