7.2 Thermodynamic Considerations

An important tool in the analysis of oxidation problems is equilibrium thermodynamics, which, although not predictive of kinetics, allows one to ascertain which reaction products are possible, whether or not significant evaporation or condensation of a given species is possible, the conditions under which a given reaction product can react with a condensed deposit, and so on. The complexity of the oxidation phenomena usually dictates that the thermodynamic analysis be represented in graphical form. The types of thermodynamic diagrams most often used in oxidation research are listed in Section 3.2 on high temperature equilibria.

The basic concepts pertinent to the construction and analysis of those thermodynamic diagrams are described in numerous thermodynamics books (Gaskell 1995; Kubaschewski and Alcock 1979).

Determination of the conditions under which a given corrosion product is likely to form is often required, e.g. in selective oxidation of alloys. In this regard, Ellingham diagrams, i.e. plots of the standard free energy of formation (ΔG°) versus temperature for the compounds of a type, e.g. oxides, sulfides, carbides, etc., are useful in that they allow comparison of the relative stabilities of each compound. In Section 3.2.1 these free energy/temperature diagrams are properly discussed; here our considerations on the thermodynamics of high temperature oxidation will finish with a brief reference to the volatility of oxides.

7.2.1 Volatility of Oxides

Some oxides exhibit high vapor pressures at very high temperatures (e.g. above 1000 °C). Oxide scales become less protective when their vapor pressures are high. Chromium, molybdenum, tungsten, vanadium, platinum, rhodium, and silicon are metals for which volatile species are important at high temperature. Vanadium is typically used in small quantities as a carbide former in alloy steels. Thus, the volatility of VO₂ is generally of no concern in oxidation of alloys.

The oxidation of Pt, and Pt‐group metals, at high temperatures is influenced by oxide volatility in that the only stable oxides are volatile. This results in a continuous mass loss. Alcock and Hooper (1960) studied the mass loss of Pt and Rh at 1400 °C as a function of oxygen pressure.

The gaseous species were identified as PtO₂ and RhO₂. These results have an extra significance because Pt and Pt–Rh wires are often used to support specimens during high temperature oxidation experiments. If these experiments involve mass change measurements, it must be recognized that there will be a mass loss associated with volatilization of oxides from the support wires.

The oxidation of pure Cr is, in principle, a simple process since a single oxide, Cr₂O₃, is observed to form. However, under uncertain exposure conditions, several complications arise, which are important both for the oxidation of pure Cr and for many important engineering alloys that rely in a protective Cr₂O₃ layer for oxidation protection. The two most important features are scale thinning by CrO₃ evaporation and scale buckling as a result of compressive stress development (Asteman et al. 1999).

The formation of CrO₃ by the reaction

7.1

becomes significant at high temperatures and high oxygen partial pressures. The evaporation of CrO₃, shown schematically in Figure 7.1, results in the continuous thinning of the protective Cr₂O₃ scale, so the diffusive transport through it is rapid. The effect of the volatilization on the oxidation kinetics has been analyzed by Tedmon (1966).

Caplan and Cohen (1961) also observed that resistance promoted volatilization of Cr₂O₃. Asteman et al. (1999) indicated that high vapor pressure of CrO₂(OH)₂ can form by reacting Cr₂O₃ with H₂O in O₂‐containing environments.

The volatilization of oxides is particularly important in the oxidation of Mo and W at high temperatures and high oxygen pressures. Unlike Cr, which develops a limiting scale thickness, complete oxide volatilization can occur in these systems. The condensed and vapor species for the Mo–O and W–O systems have been reviewed by Gulbransen and Meier (1979a, b), and the vapor species diagrams for a temperature of 1250 K are presented in Figures 7.2 and 7.3. The effects of oxide volatility on the oxidation of Mo have been observed by Gulbransen and Wysong (1948) at temperatures as low as 475 °C, and the rate of oxide evaporation above 725 °C was such that gas‐phase diffusion became the rate‐controlling process (Gulbransen et al. 1963). Naturally, under these conditions, the rate of oxidation is catastrophic. Similar behavior is observed for the oxidation of tungsten, but at higher temperatures because of the lower vapor pressures of the tungsten oxides. The oxidation behavior of tungsten has been reviewed in detail by Kofstad (1966).

The formation of SiO₂ on silicon‐containing alloys and Si‐based ceramics results in very low oxidation rates. However, this system is also one that can be influenced markedly by oxide vapor species. Whereas the oxidation of Cr is influenced by such species at high oxygen pressures, the effects for Si are important at low oxygen partial pressures. The reason for this may be seen from the volatile species diagram for the Si–O system (Rocabois et al. 1966). A significant pressure of SiO is seen to be in equilibrium with SiO₂ (s) and Si (s) at oxygen pressures near the dissociation pressure of SiO₂. This can result in a rapid flux of SiO away from the specimen surface and the subsequent formation of a non‐protective SiO₂ smoke. This formation of the SiO₂ as a smoke, rather than as a continuous layer, allows continued rapid reaction (Wagner 1958, 1965).

7.3 Kinetic Considerations

Consider the oxidation reaction

7.2

A similar reaction involving a metal phase and a gas phase was used in Section 6.8 to demonstrate the electrochemical nature of metal oxidation. Simple expressions for the growth of the MO oxide scale were also associated to support the transport of ionic and electronic species through the oxide layer. In this section, the kinetics of reaction 7.2 is discussed in more detail than it was possible in that overview chapter on high temperature electrochemistry.

The progress, W, of this reaction can be characterized using several definitions of the reaction rate. It can, for example, be defined as the rate of oxygen pickup, dn_o/Adt, where dn_o corresponds to the number of moles of oxygen consumed during time dt and A the sample area. From an experimental point of view, though, it is easier to use the sample weight gain Δm. The reaction rate can then be expressed as dΔm/Adt, where dΔm is the weight change occurring during time dt. These expressions are linked by the equation

7.3

Integration of the rate equation leads to the rate law corresponding to the corrosion process and defines the progress, W, of the reaction with time. We obtain either an implicit form:

7.4

where k is the rate constant for the reaction process, or an explicit form:

7.5

In any fundamental study of the oxidation mechanism of a metal or an alloy, one of the main factors that needs to be determined is the variation of the oxidation rate with temperature and with the pressure of the oxidizing gas.

This type of investigation has sometimes been neglected because of difficulties controlling precisely the gas pressure in thermobalances. However, for some years, it has been possible to couple such thermogravimetric equipment to devices capable of controlling and monitoring the oxygen partial pressure in a gas (e.g. an electrochemical oxygen pump, oxygen sensor). The oxygen pressure then can be precisely controlled between 1 and 10⁻²⁵ bar in gas mixtures, e.g. inert gas–oxygen, CO–CO₂, or H₂–H₂O mixtures (ASM 2003).

The rate constants of the kinetic laws often obey, under constant pressure, an Arrhenius‐type equation:

7.6

where E_a is the apparent activation energy of the process, R the gas constant, and T the absolute temperature. The apparent activation energy can be easily determined by plotting k as a function of 1/T. The slope of the straight line obtained is equal to −E_a/2.303R. A change in the activation energy could indicate a corresponding change in the limiting process for the corrosion reaction.

The main kinetic laws are of linear, parabolic, logarithmic, or cubic types, but it should be noted these are limiting cases and deviations from them are often encountered. In some cases, it is difficult, or even impossible, to obtain such simple kinetic laws from the experimental results (Evans 1960; Hauffe 1965; Kofstad 1988).

At high temperatures, the oxidation kinetics of numerous metals obey a parabolic law:

7.7

where k_p is the parabolic constant. Such a law corresponds, as will be shown later, to a corrosion rate limited by diffusion through the compact scale that is formed.

The reaction rate constant may be expressed in different units depending on the actual parameter used to define the progress of the reaction. For example, if the extent of reaction is characterized by the mass gain per unit surface area of the metal during the exposure period, t, the kinetic law is given by (Δm/A)² = k_p × t, and the rate constant is expressed in kg² m⁻⁴ s⁻¹. If the reaction rate is defined by the increase in thickness, y, of the scale, the kinetic law has the form , and the parabolic rate constant is expressed in m² s⁻¹. On the other hand, if the reaction rate is defined by the number of moles of the compound MX formed per unit area during the exposure period, t, the kinetic law has the form . In this case, the rate constant is expressed in mol² m⁻⁴ s⁻¹. There is a simple relationship between the rate constants k_p, , and :

7.8

where Ω is the molar volume of compound MX and M_x is the atomic weight of the nonmetallic element (oxygen, sulfur, etc.).

In some cases, the oxidation rate is constant, which means that the kinetic law is linear:

7.9

As will be shown later, the oxidation rate is then governed by an interfacial process such as sorption, reaction at the metal–oxide interface, etc. Using similar nomenclature as for the parabolic rate constants, k_l characterizes the reaction measured by the mass gain per unit surface area during time t and if the rate is defined by the increase in thickness of the growing scale.

The cubic law (W³ = k_ct) has been observed during the oxidation of several metals, for example, copper, nickel, and zirconium.

Logarithmic laws are observed typically in the case of many metals at low temperatures (generally below 673 K). The initial oxidation rate, corresponding to the growth of oxide layers of thickness generally less than a few tens of nanometers, is quite rapid and then drops off to low or negligible values. This behavior can be described by a direct logarithmic law:

7.10

or by an inverse logarithmic law:

7.11

The evaluation of the kinetic parameters in the case of the logarithmic law is, generally, not very precise, and this makes it difficult to validate experimentally proposed mechanisms.

The oxidation rate is frequently found to follow a combination of rate laws.

As an example, at low temperatures, a logarithmic law followed by a parabolic rate equation can be observed. At high temperature, oxidation reactions are often described by a parabolic rate equation followed by a linear law (“paralinear” regime) or a linear rate equation followed by a parabolic law (Pettit and Wagner 1964).

Typical kinetic laws characteristic of the oxidation of a large number of metals as a function of temperature were fully analyzed by Kubaschewski and Hopkins (1967), Kofstad (1988), and others. Table 7.1 gives some examples.

As an example, the kinetic laws characteristic of chromium oxidation are shown in Figure 7.4. It can be seen that at 1000 °C, a parabolic law is observed but that at 1075 °C the experimental curve is characteristic of the formation of compact scales that crack at a critical thickness due to high mechanical stresses between the metal and the growing scale; at 1200 °C, a linear rate equation is observed, with a noticeable increase in corrosion rate.

Besides the variation of the kinetic laws with temperature, a change of these rate equations with time can sometimes occur. A typical example, characteristic of the changes that may be observed as functions of temperature and time, is given in Figure 7.5. At 800 °C, for example, the following rate equations are successively observed: parabolic, paralinear, and finally linear after extended oxidation.

In many cases, it may be difficult to fit experimental data to simple rate equations, but a first approach can be to plot the W = f(t) curve using double logarithmic coordinates. In the case of a law of type Wⁿ = kt, the slope of the straight line then gives the value of n, i.e. 1, 2, and 3 for linear, parabolic, and cubic laws, respectively.

Some authors, using computer software, fit the data to a third‐degree polynomial in W:

7.12

Difficulties in evaluating the proper kinetic law are particularly important in the case of changes in the oxidation behavior during the corrosion process.

An elegant method consists of continuously monitoring the kinetic curves with exposure time and calculating, for each experimental point, the rate constant appropriate to the expected model. Deviations from this model can be readily identified (Monceau and Pieraggi 1998).

7.4 Defect Structures

The formation of an oxide scale starts with the adsorption of oxygen gas on the metal surface. During adsorption, oxygen molecules of other gaseous species in the environment dissociate and are adsorbed as atoms. These atoms initially adsorb at sites where the atom is in contact with the maximum number of surface atoms in the metal substrate. Therefore, in polycrystalline materials, grains of preferential orientation exist where the number of adsorbed atoms from the gaseous atmosphere is highest. The result of this process is a two‐dimensional adsorption layer. The presence of adsorbed layers may increase the rates of surface diffusion by orders of magnitude compared with those for surfaces with none or small amounts of adsorbate.

When the metal surface that is saturated with adsorbed oxygen atoms or atoms from other gaseous species is further exposed to the gas, the gaseous species may dissolve in the metal, and nuclei of the corrosion product are formed on the surface. These nuclei grow laterally and form a continuous film on the surface. Generally nucleation and growth of the nuclei are dependent on the composition of the substrate, the grain orientation, the temperature, and the gas partial pressure. The nuclei grow in thickness and lateral direction and the reaction rate increases with time. As soon as the nuclei impinge on each other, the growth rate decreases. Therefore, the general reaction kinetics can be described by an S‐shaped curve (Kofstad 1988).

Nuclei of all potential corrosion products can be formed on alloys initially, i.e. those that are possible from thermodynamic stability considerations. After the initial stage of oxidation, which is determined by the behavior of the nuclei, growth of the continuous scale occurs in the thickness direction. In dense oxide scales, the growth is determined by solid‐state diffusion through the scale. Corrosion products, which include the oxide scales, are ionic structures, and diffusion in such structures requires lattice disorder, i.e. the corrosion products need to be nonstoichiometric compounds. Therefore, an understanding of reaction mechanisms in high temperature conditions requires a precise knowledge of defect structures in solids. Extensive studies of defect theory have been provided by Kröger et al. (1956), Kröger (1975), Philibert (1985), Kofstad (1972), Mrowec (1980), and others, but here only an oversimplified discussion will be presented. It should also be noted that Chapter of this book gives a general treatment of lattice defects in metal compounds and includes descriptions of point defects, defect reactions, defect equilibria, and equilibrium constants. It was considered desirable to include these more general aspects so that the book could be read as a self‐contained text without the need for repeatedly consulting other books or publications.

7.4.1 Point Defects

7.4.1.1 Real Oxide Structures

A complete development of defect chemistry of inorganic compounds requires a system of notation to describe all the elements of the crystal or “structural elements,” that is, not only regular crystallographic sites but also lattice imperfections. The Kröger and Vink notation (Kröger et al. 1956), recommended by IUPAC because of its great simplicity, will be used here. Thus, in a crystal MO, a structural element of the cation sublattice has a normal charge of +2 and, consequently, an effective charge equal to zero. The electronic defects may be considered as structural elements. The electronic defect with positive charge will be written h^• (h with a superscript dot). This defect corresponds to the removal of an electron from a regular site of the cation sublattice and can also be written as (a superscript prime (′) is used for a negative charge). These rules must be followed in writing defects in equilibrium reactions:

• Electroneutrality of the total charges of the structural elements, i.e. involving normal charges, actual charges, and effective charges.
  
• Mass balance, i.e. the number of atoms of each chemical species involved in the defect reaction must be the same before and after the defect formation.
  
• Ratio of regular lattice sites, i.e. for a crystal M_pX_q: number of M sites/number of X sites should be equal to p/q.

Following these rules, it is possible to write equilibrium reactions that occur internally without involving the external environment and external equilibria involving mass exchange with the environment. The equilibrium constants will then be evaluated assuming that the activities of atoms on their normal lattice positions can be considered as unity and the activities of point defects will be approximated by their concentration, usually indicated by a double bracket [ ], and expressed as the number of moles per mole of compound.

7.4.1.2 Stoichiometry

Alkali halides, silver halides, and several oxides (Al₂O₃, MgO, etc.) are stoichiometric compounds. Some of them are characterized by vacancies and interstitials in one sublattice (e.g. AgBr, with Frenkel disorder); others possess defects in both sublattices (e.g. NaCl, with Schottky disorder). However, it is apparent that neither of these defects can be used to explain material transport during oxidation reactions, because neither defect structure provides a mechanism by which electrons may migrate.

Considering a diagrammatic representation of the oxidation process shown in Figure 7.6, it is seen that either neutral atoms or ions and electrons must migrate in order for the reaction to proceed. In these cases, the transport step of the reaction mechanism links the two phase‐boundary reactions as indicated. There is an important distinction between scale growth by cation migration and scale growth by anion migration in that cation migration leads to scale formation at the scale/gas interface, whereas anion migration leads to scale formation at the metal/scale interface.

In order to explain simultaneous migration of ions and electrons, it is necessary to assume that the oxides that are formed during oxidation are nonstoichiometric compounds. From a macroscopic viewpoint, two alternative classes of nonstoichiometric compounds can be considered:

1. M_aX_b compounds for which . These compounds are ionic semiconductors with metal excess (M_a+δX_b, where δ is the deviation from stoichiometry in comparison with the stoichiometric composition M_aX_b) or with nonmetal deficit (M_aX_b−δ). ZnO is a typical n‐type semiconductor with a metal excess; TiO₂, Nb₂O₅, MoO₃, and WO₃ are typical n‐type semiconductors with nonmetal deficit.
   
2. M_aX_b compounds for which . These include the p‐type semiconductors with metal deficit (FeO, NiO, Cr₂O₃, Al₂O₃) or with an excess of nonmetal (UO₂).

In order to allow extra metal in ZnO, it is necessary to postulate the existence of interstitial cations with an equivalent number of electrons in the conduction band. The structure may be represented as shown in Figure 7.7. Here, both Zn⁺ and Zn²⁺ are represented as possible occupiers of interstitial sites. Cation conduction occurs over interstitial sites, and electrical conductance occurs by virtue of having the “excess” electrons excited into the conduction band. These, therefore, are called “excess” or “quasi‐free” electrons.

The formation of this defect may be visualized, conveniently, as being formed from a perfect ZnO crystal by losing oxygen: the remaining unpartnered Zn²⁺ leaving the cation lattice and entering interstitial sites and the two negative charges of the oxygen ion entering the conduction band. In this way, one unit of ZnO crystals is destroyed, and the formation of the defect may be represented by

7.13

for the formation of , doubly charged Zn interstitials, or

7.14

for the formation of , singly charged Zn interstitials.

The two equilibria shown above will yield to thermodynamic treatment, giving Eq. 7.15 for the equilibrium in Eq. 7.13:

7.15

or, since the defects are in very dilute solution, we may assume that they are in the range obeying Henry’s law, when the equilibrium may be written in terms of concentrations and [e′] as in Eq. 7.16:

7.16

If Eq. 7.13 represents the only mechanism by which defects are created in ZnO, then Eq. 7.17 follows:

7.17

Hence, putting Eq. 7.17 into Eq. 7.16, we obtain Eq. 7.18,

7.18

or Eq. 7.19, and therefore we obtain Eq. 7.20:

7.19

7.20

Similarly, applying the same analysis to the reaction shown in Eq. 7.14, the result shown in Eq. 7.21 is obtained:

7.21

Measurement of electrical conductivity as a function of oxygen partial pressure carried out between 500 and 700 °C (von Baumbach and Wagner 1933) indicated that the conductivity of ZnO varied with oxygen partial pressure having exponents between 1/4.5 and 1/5. This result indicates that neither defect mechanism predominates and the actual structure could involve both singly and doubly charged interstitial cations (Kofstad 1972).

Similar approaches can be applied to nonstoichiometric compounds with cation vacancies (Cu_2−δO‐type oxide), oxygen interstitials (UO_2+δ‐type oxide), etc. (Farhi and Petot‐Ervas 1978; Pope and Birks 1977).

7.4.1.3 Mass and Electrical Transport

Intragranular or volume diffusion in crystalline compounds takes place through crystal imperfections and mainly through the movement of point defects. Several types of mechanisms may be considered, as shown schematically in Figure 7.8, but mass transport generally occurs by hopping mechanisms from a well‐defined site of the crystal into another adjacent site.

Consider a one‐dimensional flux of particles (atoms, ions, point defects, or electrons) in the Ox direction. Let C (x, t) be the defect concentrations (number of particles per unit volume) at the coordinate x and at time t. In a chemical potential gradient and without an electrical potential gradient or other type of driving force, a flux, J, of particles occurs in the Ox direction:

7.22

where D is the diffusion coefficient of the particle.

Under an additional electrical potential gradient, the particle flux would obey the following general equation:

7.23

where , termed the electrochemical potential, is related to the chemical potential μ by the equation

7.24

where z is the particle charge number, F the Faraday constant, φ the electrical potential, and ±zF the electrical charge.

If the ion movements within one sublattice of the binary compound M_aX_b produce displacements of ions only in that sublattice, M or X diffusion is termed self‐diffusion, and the self‐diffusion coefficient D_j of component j will obey the following equation:

7.25

where C_j is the concentration of component j.

Let J_δ be the flux of the defect, δ, in the j sublattice at the coordinate x and in the absence of an electric field. The diffusion coefficient D_δ of the defect δ is defined by equation

7.26

where C_δ is the concentration of the defect δ.

As a general rule, the relationship between D_j and D_δ may be written as

7.27

If N_δ is the mole fraction of defects δ in the j sublattice, we may write

7.28

This relationship shows that the self‐diffusion of component j is proportional to the mole fraction of defect δ contained in the j sublattice.

A comparison of the self‐diffusion coefficients of anions and of cations may allow identification of the component that provides the majority of mass transport within the crystal. Thus, nonstoichiometric oxides such as NiO, FeO, and Cu₂O contain metal vacancies; this observation is in agreement with the order of magnitude of the diffusion coefficients, i.e. D_M > D_O (D_M/D_O ≈ 10²–10⁴). In contrast, in some oxides such as TiO₂ that have an oxygen deficit, it has been observed that D_O ≈ D_Ti. This result is not in contradiction with the assumption of several authors who postulate that both oxygen vacancies and titanium interstitials are simultaneously present in this oxide.

Electrical transport in ionic compounds does not necessarily occur by means of point defects. Electrical conductivity due to a charge carrier is given by

7.29

where C is the molar concentration of charge carriers per unit volume and U is the electrical mobility (expressed in m² s⁻¹ V⁻¹). The mobility U corresponds to the velocity of the charged particles under an electric field equal to unity.

In ionic crystals, the total conductivity σ_t is generally written in terms of ionic and electronic conductivities as

7.30

Let σ_δ be the partial conductivity relevant to the defect δ, and considering the definition of the electrical mobility U_δ, we obtain

7.31

If σ_j is the contribution to the total conductivity of the charged species j, we can write

7.32

Since

7.33

we obtain the Nernst–Einstein equation:

7.34

In this equation, σ_j is the ionic contribution of species of type j to the total conductivity, D_j is the self‐diffusion coefficient of particles j, and C_j is the volume concentration of regular sites in the sublattice that contain species j.

Since the mobility of electronic defects is much higher than that of point defects (U_ε > U_δ), it can be said that the total electrical conductivity is essentially electronic. Also, it is easy to show that the conductivity varies with oxygen pressure in the same way as does the concentration of the predominant ionized defect (although the current is carried by electronic defects).

The temperature dependence of the conductivity is determined by both the charge carrier mobility and concentration terms.

When ion movements involve jumps between definite sites of the crystal, an energy barrier ΔG_m has to be overcome. The defect mobility then increases strongly with temperature according to an exponential law (activated process):

7.35

where ΔG_m is the free energy of migration of the defect.

The temperature dependence of electron mobility is a function of the electronic structure of the crystal. The electron movement is an activated process, and the electronic mobility obeys the following equation:

7.36

where E_ε is the overall activation energy for polaron migration in the periodic field within the crystal or, in other words, for polaron (electron and distortion field) scattering by lattice vibrations and/or imperfections, also known as polaron hopping and usually treated as a diffusion process.

The determination of the variation of the electronic mobility with temperature may allow us to identify the migration mechanism of electrons in the lattice. Whatever the nature of the charge carrier (ionic or electronic defects), the concentration increases with temperature according to

7.37

where ΔG_f is the free energy of formation of defects.

Whatever the nature of the conduction mechanism, the electrical conductivity is proportional to the product of the drift mobility and the charge carrier concentration, which varies exponentially with temperature. The crystal conductivity always increases with increasing temperature due to the exponential increase in the number of charge carriers. This characteristic differentiates covalent/ionic compounds from metallic conductors, which exhibit a decrease of the electrical conductivity with increasing temperature (Kröger 1975; Mrowec 1980; Philibert 1985).

7.4.2 Line and Planar Defects

As diffusion along line and surface defects, including dislocations, grain boundaries, internal and external surfaces, etc., is generally more rapid than lattice diffusion, they are termed high‐diffusivity or easy diffusion paths. This type of diffusion is often called short‐circuit diffusion. The contribution of grain boundary diffusion to the total diffusion flux decreases as the temperature increases for two main reasons:

• The grain size is larger and the temperature is higher because of grain growth.
  
• The activation energy for grain boundary diffusion is less than that for intracrystalline diffusion (from 20% to 30% smaller).

The effective diffusion coefficient may be defined by the Hart equation:

7.38

where f is the volume fraction of short‐circuit paths, D_v the lattice diffusion coefficient, and D_gb the short‐circuit diffusion coefficient. D_eff may be identified with D_v, the intracrystalline diffusion coefficient, at high temperatures, but at low temperatures, the short‐circuit contribution to diffusion can become significant. In general, in accordance with Tamann’s empirical law, grain boundary diffusion would be expected to dominate at lower temperatures, say, below a transition temperature of between 1/2 and 2/3 of the absolute melting temperature of the crystal. Conversely, the contribution made by short‐circuit diffusion processes will be negligible at higher temperatures. Grain boundary diffusion in growing oxide scales has been reported for NiO (Atkinson et al. 1986), Cr₂O₃ (Hussey and Graham 1996), Al₂O₃ (Prescott and Graham 1992), and other product films.

7.4.3 Three‐Dimensional Defects

Stress generation in the oxide layer and the underlying metal may cause through‐scale cracking, spalling of the oxide, stratification phenomena, or even detachment of the scale. These phenomena lead to loss of protective properties and faster degradation of metals and alloys. The sources of stress may be either internal (scale growth) or external (mechanical and/or thermal stresses). Often, due to mechanical stresses, a porous layer may develop after the oxide scale has reached a critical thickness. The two main sources of stress are growth stresses, which develop during isothermal formation of the scale, and thermal stresses, which arise from differential thermal expansion between the oxide scale and the metal or alloy during temperature changes.

7.4.3.1 Growth Stresses

Observed stresses depend on the oxidation mechanism and on the physicochemical properties of the alloy and of the oxide. The most important causes of growth stresses are (Béranger et al. 1987) the volume difference between the oxide and the metal (Pilling and Bedworth rule), the oxidation mechanism (for example, internal or external oxidation of alloys), oxygen dissolution in alloys, epitaxial constraints, physicochemical changes in the alloy or scale during the growth, specimen geometry, etc.

Two different types of growth stresses can be distinguished: geometrically induced growth stresses caused by the surface curvature of components and the intrinsic growth stresses. As can often be seen in oxidation experiments, the oxide scales crack at the edges of the specimens, initially leading to a locally increased attack at these sites. Such cracking is usually due to geometrically induced growth stresses that arise at edges and corners due to the small surface curvature radius. This situation has been dealt with quantitatively by Manning (1981).

With the help of models, the tangential and radial stresses can be calculated for the ideal case of curved surfaces with a constant radius of curvature. Introduction of the oxide displacement vector M is helpful here; M lies perpendicular to the oxide–metal interface and describes the displacement of a reference point in the film resulting from the oxidation. The magnitude and sign of M are incorporated in M, which is calculated as follows:

7.39

where a is the fraction of oxide formed on the scale surface, (1 − a) is the fraction of oxide formed at the metal–oxide interface, V is the volume fraction of metal consumed in the oxidation by injecting vacancies into the metal, (1 − V) is the volume fraction of metal consumed in the oxidation that originates directly from the metal surface, and PBR is the Pilling–Bedworth ratio (see later in this section).

Oxidation leads to an increase in the strain in the circumferential direction (tangential strain ) with a rate of

7.40

where R_s is the radius of curvature of the surface (concave R_s < O, convex R_s > O) and h is the metal recession (increase in oxide film thickness dx = PBR dh). Equation 7.40 allows the tangential stresses to be calculated assuming linear elastic behavior. The magnitude of the maximum radial stresses, , is given by

7.41

The relationship between the signs of the tangential and radial strains and stresses in the scale and at the metal–oxide interface, respectively, are

7.42

7.43

A plus sign indicates tensile stress, and a minus sign means compressive stress. The sign and level of the stresses in the scale depend on its growth direction and on the radius of service curvature, as well as the PBR. The latter was introduced in 1925 in order to explain the formation of growth stresses during oxidation and describes the volume change that is involved in the transition from the metal lattice to the cation lattice of the oxide when only the oxygen anions are diffusing. In other words, the PBR corresponds to the ratio of the volume per metal ion in the oxide to the volume per metal atom in the metal (Fromm 1998):

7.44

It was argued that if the PBR was less than 1, the growth stresses would be tensile and the oxide would crack and not cover the entire metal surface. As indicated in Table 7.2, alkali and alkaline earth metals belong to this class of materials. On the other hand, if the PBR was higher than 1, compressive stresses would develop and the oxide could be protective, at least during the early stages of oxidation. The majority of metals fall into this category.

Oxide	K2O	CaO	MgO	CeO2	Na2O	CdO	Al2O3	ZnO	ZrO2
PBR	0.45	0.64	0.8	0.90	0.97	1.21	1.28	1.55	1.56
Oxide	Cu2O	NiO	FeO	TiO2	CoO	SiO2	Cr2O3	Ta2O5	Nb2O5
PBR	1.64	1.65	1.7	1.73	1.86	1.9	2.07	2.5	2.7

We now know that the Pilling–Bedworth rule regarding protective behavior exhibits several exceptions. Important examples are tantalum or niobium where, even though the PBR is substantially larger than unity, cracks develop in the oxide scale after extended exposure and these produce non‐protective conditions. Whereas the Pilling–Bedworth paper (1923) was a significant advance at the time, it is now recognized that the approach was incomplete and that the influence of the difference between the molar volume of metal and oxide depends on the oxide growth mechanism. However, the Pilling–Bedworth approach may be of great help for the assessment of the geometrically induced growth stresses, as shown earlier.

7.4.3.2 Thermal Stresses

In most applications, high temperature alloys are subjected to temperature fluctuations even under nominally isothermal conditions. In this case, though, the resultant stresses in the oxide layer, resulting from the difference in the coefficient of thermal expansion (CTE) of the metal and oxide (see Table 7.3), are small and may be neglected. This will not be the case, however, for large thermal cycles or during cooling to room temperature when large stresses, perhaps of 1 GPa order, are produced in the oxide layer. Metals have, generally, a higher CTE than oxides (Table 7.3), and consequently, tensile stresses are induced in the oxide scale on heating and compressive stresses during cooling.

System	Oxide: 106·αox	Metal: 106·αM	Ratio: αM/αox
Fe/FeO	12.2	15.3	1.25
Fe/Fe2O3	14.9	15.3	1.03
Ni/NiO	17.1	17.3	1.03
Co/CoO	15.0	14.0	0.93
Cr/Cr2O3	7.3	9.5	1.30
Cu/Cu2O	4.3	18.6	4.32
Cu/CuO	9.3	18.6	2.0

The thermally induced stresses can be calculated from the CTE according to the following equation (Cathcart 1975):

7.45

where α is the CTE for the metal and the oxide, E is Young’s modulus for the metal and the oxide, d is the thickness for the metal and the oxide, and υ is Poisson’s ratio. ΔT stands for the temperature change. The CTEs for technical materials can be found in many of the materials producers’ brochures, and those for corrosion products are given in the literature (Schütze 1997). In most cases, the CTEs can be approximated by linear behavior in the temperature range concerned, but in some cases, where phase changes occur in the scale during the temperature change, nonlinear temperature dependence is found for the CTE. This is, for example, the case for several sulfide layers (Schulte and Schütze 1999) and is particularly important for magnetite and some iron‐based spinels (Armilt et al. 1978), affecting stresses in oxide scales on low‐alloy steels. This naturally decisively affects the stress situation in the oxide scales on low‐alloy steels (Christl et al. 1989). In the temperature range between about 600 and 450 °C, the magnetite partial layer is under tensile stress when cooling from 600 °C. At lower temperatures, this oxide partial layer may come under compressive stresses, depending on the metallic substrate and its CTE. In the hematite layer, the stresses are always compressive, as the CTE always lies below that of the low‐alloy steel (the exception is 9% chromium steel at temperatures below 150 °C).

7.4.3.3 Mechanical Scale Failure

Growth stresses and thermal stresses may be relieved through various mechanisms that could operate simultaneously:

• Plastic deformation of the oxide scale.
  
• Plastic deformation of the metal substrate.
  
• Spalling of the oxide from the alloy.
  
• Cracking of the scale.

When plastic deformation is not sufficient for stress relief, cracking may develop in the scale. It is the more efficient relaxation mechanism but will result in a sudden increase in corrosion rate. The metal oxidation may exhibit repeated regular sequences of cracking and healing of the scale.

Under tensile stresses (heating to temperatures higher than the oxidation temperature or over convex regions of a nonplanar surface), cracks appear as soon as the elastic fracture strain is reached. This critical value will be significantly less than 1% even at high temperatures.

Under compressive stresses, the degradation leads to spallation and the mechanisms are more complex. Two processes are necessary to produce spalling: transverse cracking through the oxide and decohesion along the metal–oxide interface. Two routes of spallation have been identified: the case corresponding to a low cohesive strength of the oxide and a high adhesive strength of the scale on the substrate surface (route 1: cracking of the oxide before decohesion) and the case corresponding to a high cohesive strength of the oxide and poor adhesion of the oxide to the metal (route 2: decohesion before metal cracking). Figure 7.9 illustrates these two distinct mechanisms.

7.5 Compact Scale Growth

7.5.1 Elementary Chemical Steps

The overall oxidation reaction of a metal M may be written as shown in Eq. (6.41) (Section 6.8). The reaction can proceed only if diffusion of matter (oxygen or metal) occurs through the solid scale M_aO_b. If the scale is porous, mass transport occurs by oxygen diffusion; if the scale is compact, mass transport occurs by means of solid‐state diffusion. In the latter case, the oxidation mechanism consists of at least four steps (Barret 1975; Mrowec and Stoklosa 1971; Yang et al. 1998):

• Surface step – Oxygen adsorption on the oxide.
  
• External step – Matter exchange at the adsorbed phase/oxide interphase.
  
• Diffusion step – Ionic transport through the oxide scale.
  
• Internal step – Matter exchange at the oxide–metal interface.

Of course, mass transport by migration of ionized point defects is accompanied by simultaneous electrical transport, which complicates the process. In this section, the kinetics of the oxidation process leading to the growth of a compact scale are analyzed.

Let a chemical elementary step be a local reversible reaction that occurs without the formation of a distinct intermediate product, i.e. the reaction proceeds in a single step, and let an interphase elementary step be a chemical process involving matter exchange between two distinct phases. The main difficulty encountered for the formulation of these steps relates to the conditions required for the application of classical theories (Eyring theory) for the calculation of the step rates (Glasstone et al. 1941).

We will have to choose, for the adsorbed phase as well as for the oxide, structural models that exhibit ideal behavior for the reacting species in their own phase. Moreover, we will assume that the theory of absolute rates can, under these conditions, be extended to heterogeneous elementary steps involving matter exchange between two different phases.

On a solid surface, the atomic environment is modified in comparison with that in the bulk. The resulting imbalance of the forces in the surface of solids produces attractive forces for gas molecules or atoms. The phenomenon of adsorption can then produce an excess of gas atoms or molecules on the surface compared with the concentration in the adjacent gas phase. For adsorption to occur spontaneously, the process must produce a decrease in free energy, but since that is also a decrease in system entropy, adsorption is always an exothermic process. Consequently, the amount of adsorbed gas at equilibrium at constant pressure (the adsorption isobar) decreases with increasing temperature.

Depending on the nature of the forces involved, adsorption processes may be classified as physical adsorption (also termed van der Waals adsorption or physisorption) or chemical adsorption, usually abbreviated to chemisorption (Brunauer et al. 1938; Langmuir 1918). Physisorption is generally quasi‐instantaneous, while chemisorption often proceeds slowly, involving an activation energy E_a. Thus, the chemical adsorption rate becomes appreciable only at sufficiently high temperature.

Many theories and models have been proposed to explain the shape of adsorption isotherms that represent the variation of adsorbed volume as a function of gas pressure or of the p/p₀ ratio (p₀ is the saturation vapor pressure at the experimental temperature). The description of monolayer adsorption can be made using as variable the fraction of the available adsorption sites that are occupied by adsorbed atoms or molecules, θ = s/s₀, where s₀ is the number of adsorption sites that are initially available per unit surface area and s the number of occupied surface sites per surface area unit (thus, θ is the fraction of occupied sites).

Chemisorption involves partial electronic transfer between adsorbed molecules and the substrate. The solid surface appears inhomogeneous and exhibits specific “active” sites on which chemisorption takes place preferentially. When temperature is increased, the amount of adsorbed gas by chemisorption increases because it is an activated process; then the adsorption isobar passes through a maximum because chemisorption is an exothermic process. Since the establishment of high temperature oxidation requires at least the presence of one monolayer on the surface, it can be assumed that a chemisorbed phase is produced as a surface step.

In this discussion, this adsorbed phase is considered as a two‐dimensional solution (i.e. sorption of a monolayer) of free surface sites, s, and of occupied sites, leading to the formation of a superficial compound referred to as O‐s (atomically chemisorbed oxygen) where O is a particle of the gas phase (O₂). Such a solution may be considered as ideal since it is assumed that no interaction occurs between the free and occupied sites. The sorption process may then be described by an equation representing the balance between two opposite reactions with rate constants and , respectively:

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