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).

7.4.2 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.

7.5.1 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:

7.46

with denoting the equilibrium constant of the elementary step of sorption.

Under these conditions, if p denotes the partial pressure of the species, θ^eq the fractional surface coverage, the law of mass action applied to the equilibrium leads to

7.47

indicating that the sorption is governed by the Langmuir equation (dissociation occurs on chemisorption, which is likely at high temperature). If dn_O‐s is the number of O‐s particles formed by adsorption per unit area during the time dt, the rate of the sorption process can be given by the equations

7.48

or

7.49

with

7.50

The proposed model to account for the external and internal steps uses the concept of point defects, and, for simplicity, it can be assumed that only one defect is predominant in the lattice, that is, either the metal vacancy or the oxygen interstitial for the case of a p‐type semiconductor and either the oxygen vacancy or the metal interstitial for the case of an n‐type semiconductor. The defects are then generated at the external interphase (cationic diffusion, p‐type semiconductor) or at the internal interphase (anionic diffusion, n‐type semiconductor) (see Figure 7.6).

Then, in the case of an n‐type semiconductor with metal vacancies, the external step can be described by

7.51

with denoting the equilibrium constant of the external step of matter exchange. The external reaction rate is given by

7.52

Under equilibrium conditions, for a given value of θ, we have

7.53

with

7.54

Again for a p‐type semiconductor with metal vacancies, we have for the internal step

7.55

7.56

with

7.57

In a thick, growing oxide scale, a positive space charge can appear in contact with the metal balanced by a negative space charge localized near the external interphase. This charge distribution induces, at any point within the scale, an electric field that both accelerates the positively charged ionic defects and slows down the negatively charged electronic defects until no net electrical current flows through the scale. Consequently, stationary concentration profiles are established in the MO scale, the electrically neutral zone extending over practically all of the scale thickness (Wagner 1933a,b).

Taking into account the possible electrical potential gradient through the scale, the particle fluxes are given by

7.58

7.59

In these expressions, C_δ (C_ε) is the volume concentration of point (electronic) defects with z degree of ionization, D_δ (D_ε) is the diffusion coefficient, U_δ (U_ε) is the electric mobility of these charge carriers, and φ is the electric potential at the coordinate x. C_δ and [δ] are linked by the relationship C_δ = [δ]/Ω, Ω being the volume of one mole of oxide. A positive sign (+) has to be used if the defects are positively charged and a negative sign (−) for negatively charged defects.

A brief kinetic analysis of the four elementary steps for reaction (6.41) has been presented in this section. The rate expressions of the elementary steps were expressed in a form involving the deviation from equilibrium, i.e. within the framework of the thermodynamics of irreversible processes. The general system of equations relating to the growth of an oxide scale MO can then be established by expressing the mass balance at both sides of each interphase in the adsorbed phase and within the scale. These equations being differential cannot be solved analytically. However, it is possible to make the simplifying assumption that the concentrations tend to become time independent, i.e. a quasi‐steady state develops. On this basis, the system of differential equations allowing calculation of the reaction rate can be solved analytically, leading to considerable simplifications. It should be recalled that if one of the rate constants or the diffusion coefficient has a finite value, all the other constants having very large values, we deal with what is called a pure regime (Gesmundo and Viani 1981; Graham et al. 1972). In all other cases, it is called a mixed regime (Deal and Grove 1965; Pettit and Wagner 1964). Hereinafter, we describe pure diffusional regimes.

7.5.2 Diffusion‐Controlled Oxidation

Assuming that cationic transport across the growing oxide layer controls the rate of scaling and that thermodynamic equilibrium is established at each interphase, the outward cation flux, , is equal and opposite to the inward flux of cation vacancies, and we can write

7.60

where y is the oxide thickness, is the diffusion coefficient for cation vacancies, and and are the vacancy concentrations at the scale/metal and scale/gas interphases, respectively.

Since there is thermodynamic equilibrium at each interphase, the volume of is constant, and we have

7.61

where

7.62

Integrating and noting that y = 0 at t = 0, we obtain

7.63

which is the common parabolic rate law. Furthermore, since it has been shown that the cation vacancy concentration is related to the oxygen partial pressure by the equation

7.64

the variation of the parabolic rate constant with oxygen partial pressure can be predicted by

7.65

and since is usually negligible compared with , we have

7.66

Clearly, the concentration gradient in the scale never equals zero, and, therefore, scale growth never stops.

Originally, Eq. 7.63 was derived by Wagner in a theoretical detailed analysis of the electrochemical potential situation and the transport conditions in the scale. Figure 7.10 gives a summary of the conditions for which the theory is valid.

Assumptions are as follows:

• The oxide layer is a compact, perfectly adherent scale.
  
• Migration of ions or electrons across the scale is the rate‐controlling process.
  
• Thermodynamic equilibrium is established at both the metal/scale and scale/gas interfaces.
  
• The oxide scale shows only small deviations from stoichiometry, and, hence, the ionic fluxes are independent of position within the scale.
  
• Thermodynamic equilibrium is established locally throughout the scale.
  
• The scale is thick compared with the distances over which space charge effects (electrical double layer) occur.
  
• Oxygen solubility in the metal may be neglected.

This model led to the final equation of the parabolic rate constant, which is (Kofstad 1988; Wagner 1975)

7.67

where , D_M, and D_O are given in units of cm² s⁻¹, with D_M and D_O the self‐diffusion coefficients for random diffusion of the respective ions (metal and oxygen) and z the valence of the respective ion (anion and cation).

The parabolic rate law that was derived for thickness growth can also be modified for weight gain by oxidation. In this case, has to be replaced by according to the equation

7.68

Values of ∫_z are given in Table 7.4 for several oxides. These values are, however, based on the assumption that the scale is free of pores and cavities and consists of only one phase. Under practical conditions, this is not usually the case, and therefore these values should only be used as estimations.

The oxidation rate constant K_p is the most important parameter for describing oxidation resistance. If K_p is low, the overall oxidation rate is low and metal consumption occurs at a very low rate. This is typical for protective oxidation. If K_p is high, metal consumption occurs at a high rate and the case of non‐protective oxidation exists. From K_p the metal consumption rate can also be calculated (Heitz et al. 1992). This requires knowledge of the stoichiometry of the oxide and the specific weight values, as well as the molar weights of the reactants. Then the metal consumption can be calculated.

The real value of Wagner’s analysis lies in providing a complete mechanistic understanding of the process of high temperature oxidation under the conditions set out. The predictions of Wagner’s theory for n‐type and p‐type oxides have been extensively examined by several workers (Heitz et al. 1992; Mrowec and Przybylski 1977a, b; Sarrazin and Besson 1973). For many systems, the obtained rate constants are generally several orders of magnitude larger than those that one would calculate from lattice diffusion data from Eq. 7.67. This discrepancy indicates that “short‐circuit transport” is contributing to growth of the oxide film.

7.5.3 Short‐Circuit Diffusion

Lattice diffusion is the dominant process of mass transport at high temperature, provided there is a sufficiently high defect concentration, but mass transport can also occur along dislocations or grain boundaries. Thus, the overall transport in polycrystalline oxide scales generally results from two fluxes of matter in parallel: an intragranular flux J_V and a flux along grain boundaries J_gb.

The oxidation rate may be expressed as a function of the effective diffusion coefficient defined by Eq. 7.38 (see Section 7.4.1). The scale growth rate obeys the differential equation

7.69

where ΔC is the defect concentration difference through the oxide scale and Ω the molar volume of the oxide. Integration of Eq. 7.69 leads to

7.70

where is the effective parabolic constant.

Assuming that the oxide is dense and pure and the grains are spherical, the following rate equation can be obtained (Caplan et al. 1972; Graham et al. 1972; Khoi et al. 1975):

7.71

where R₀ is the initial grain radius and K is a constant, which is a function of, among others, the surface energy of grain boundaries and their thickness and of the diffusion coefficient of matter across the grain boundaries. Further numerical development of this simple model has allowed a grain size distribution, different grain boundary widths, or different laws of grain growth to be taken into consideration.

7.6.1 Compact Subscales

The theory of multilayered scale growth on pure metals has been treated by Yurek et al. (1974). The hypothetical system treated is shown in Figure 7.11. It is assumed that the growth of both scales is diffusion controlled with the outward migration of cations large relative to the inward migration of anions. The flux of cations in each oxide is assumed to be independent of distance. Each oxide exhibits predominantly electronic conductivity, and local equilibrium exists at the phase boundaries. The total oxidation reaction is

7.72

The cation vacancies are assumed to be neutral. The cation flux in subscale (1) is

7.73

and the amounts, Q, of metal consumed per unit area to form layers (1) and (2) are, respectively,

7.74

The fractions of the cation flux involved in the growth of subscales (1) and (2) are, respectively.

7.75

7.76

with .

With this model, it is not possible to express simply the ratio of the thickness of both subscales. However, the ratio of parabolic rate constants can be obtained as

7.77

When one of the layers is much larger than the other, this expression simplifies to

7.78

for example, if y₂ ≪ y₁. It appears that this ratio is directly proportional to the ratio of parabolic rate constants for the growth of each layer alone, i.e. roughly proportional to the ratio of the self‐diffusion coefficients of the mobile species.

This theory has been shown by Garnaud (1977) to describe the growth of CuO and Cu₂O on Cu, by Garnaud and Rapp (1977) to describe the growth of Fe₃O₄ and FeO on Fe, and by Hsu and Yurek (1982) to describe the growth of Co₃O₄ and CoO on Co.

7.6.2 Paralinear Oxidation

Generally, experience shows that the oxidation of a metal is often protective during the early stages but that the protective properties of the scale can be partially or totally lost during later stages. For example, according to the Haycock–Loriers model (Haycock 1959; Loriers 1950), the oxidation process involves the concurrent growth of an inner compact layer of MO, controlled by a diffusion mechanism, and its progressive transformation at its outer interface into an external porous layer MO′. The rate of growth of the inner compact layer controlled by diffusional transport is given by

7.79

where Δm₁ is the mass of oxygen in the compact scale, K_p is the parabolic rate constant for growth of the layer, and K₁ is the rate constant for its transformation. The growth rate of the outer porous layer controlled by the reaction at its interface with the inner compact layer is given by

7.80

where f is the ratio of the oxygen content in the oxide MO′ to that in the oxide MO. If the scale consists of two layers of the same oxide, f = 1, and the weight gain Δm is given by the rate equation

7.81

where the rate constant K₁ is characteristic of a transformation in the solid state that does not depend on the oxygen pressure. The function Δm = Δm₁ + Δm₂ = F(t) can be approximated by the equation

7.82

which describes paralinear oxidation for which parabolic kinetics predominate during the early stages of oxidation becoming linear at longer times. This is illustrated in Figure 7.12. Paralinear oxidation is observed during the oxidation of a wide range of metals, especially if mechanical damage occurs to the scale during thermal cycling (Chang et al. 1990; Rapp and Colson 1966). This is more likely to be the case if the CTE of the oxide is much less or greater than unity. The simultaneous oxidation and evaporation leads to the formation of porous and partially porous scales (Tedmon 1966).

7.6.3 Stratified Scales

Several metals, particularly in columns IV and V (Ti, Zr, Nb, Ta) of the periodic table, form stratified scales during their oxidation at high temperatures as a result of periodic cracking of the growing oxide. From a kinetic point of view, such cracking leads to two types of rate laws (Figure 7.13): the first one is described by successive parabolic or cubic periods (type 1), and the second one by an initial parabolic or cubic period followed by a near‐linear period.

Examination of the scale formed on such specimens, either on fractured or polished cross sections, shows that for both type 1 and type 2 rate laws,

• A compact and adherent scale is formed before the first transition.
  
• After the first transition, a porous scale exists that consists of well‐defined layers formed essentially parallel to the metal surface; these may be separated by isolated cracks, also parallel to the metal surface, or by connected pores.

Mass transport through the oxide occurs by oxygen, mainly by vacancies V₀ (diffusing from the internal to the external interface). Since the scale grows in a confined space at the internal interface, the increase in volume on oxide formation, associated with a PBR greater than unity, may generate large compressive stresses in the oxide (several gigapascals have been measured). The metal is concurrently submitted to tensile stresses of smaller magnitude (several tens of megapascals) due to its larger thickness. The high compressive stresses in the oxide are probably responsible for the observed tendency for the kinetics to approach a cubic rate law due to an associated reduction in the oxygen diffusion flux. In all cases, oxygen pressure has no influence on the rate constants.

As oxide thickness increases, compressive stresses may also increase and result in localized spallation and/or cracking (Stringer 1960a, b). Then, free access of the gas to a bare metal surface occurs and reoxidation results. Two cases can be envisaged. In the first, the bare metal is unchanged compared with that at the beginning of oxidation, in terms of composition, microstructure, or mechanical properties. Reoxidation occurs exactly as in the initial period, and the rate law is the exact repetition of the first pre‐transition curve. Kinetics of type 1 (Figure 7.13) are then observed with a succession of parabolic‐type periods.

In the second example, the bare metal differs from the initial state, for example, dissolution of oxygen may have occurred, leading to increased hardness and lower creep relaxation rates. The second oxide layer that forms does not then reach the thickness of the first since stress increases at a higher rate and early separation occurs from the metal. In the case of TiO₂ growth on titanium at 850 °C, for example, the second and all subsequent layers have a thickness of 1–2 µm, whereas the first attains 10–15 μm. The spallation or cracking of these scales does not occur at the same time for all locations on the specimen surface, and the resulting law is approximately linear with an increased rate compared with that in the pre‐transition period. The system can be described as a metal covered by an oxide of statistically constant thickness (Munir and Cubicciotti 1983; Newcomb and Bennett 1993).

7.7.1 Atmosphere Control

When a process is considered for use in industry, it must be effective, reliable, and economical. For most applications these requirements rule out systems based on high vacuum, and the general practice is to use atmospheres derived from fuels. The gases used are therefore mixtures of N₂, CO, CO₂, H₂, H₂O, and CH₄, which make up the products of combustion of fuels. More recently, atmospheres based on nitrogen have been increasingly used.

Starting from the fuel and air, various types of atmospheres can be produced. The main, or common, differentiation is between “exothermic” and “endothermic” atmospheres. The nomenclature is ambiguous and it is as well to be clear about its meaning. An exothermic atmosphere is produced exothermically by burning the fuel with measured amounts of air. This type of atmosphere has the highest oxygen potential. An endothermic atmosphere is produced by heating, by external means, a mixture of fuel gas with air over a catalyst to provide a gas containing reducing species. This atmosphere has a low oxygen potential, and heat is absorbed during its preparation; hence the atmosphere is described as endothermic. A glance at the analysis of dried, stripped, exothermic atmospheres will confirm that they are predominantly pure nitrogen. Basically, the fuel has been used to remove oxygen from the air.

The economics of using nitrogen as a controlled‐atmosphere source become more attractive when such factors as safety, reliability, and productivity are considered. Furthermore, the present tendency is to move away from oil toward electricity, in which case nitrogen atmospheres will be particularly attractive.

Modern techniques are currently available using carburizing and nitriding systems under vacuum. In these processes of vacuum carburizing and plasma carburizing, the components are heated under vacuum to around 950 °C. Methane is leaked into the chamber to a pressure of between 3 and 30 mbar to add carbon to the system. In the absence of a plasma, the methane will only decompose to the extent of about 3%, probably on the surface of the components according to a sequence such as that shown in the following equation:

7.83

These reactions may be stimulated to provide 80% decomposition by using a plasma process to excite the methane molecule. In this case, the molecular breakdown may occur in the plasma to produce charged species. Hydrocarbons other than methane may be used as the feedstock. The usual operating sequence involves flushing and evacuation, heating to a temperature under the inert atmosphere, and carburizing for a predetermined time followed by a diffusion anneal in a carbon‐free atmosphere. This cycle is designed to provide optimum surface carbon content and carburized depth (Birks et al. 2006; Fromm 1998; Grabke and Meadowcroft 1995).

The main application of controlled atmospheres is in the area of heat treatment of finished machined components or of articles of complex shape, which cannot easily be treated subsequently for the removal of surface damage. In this context, the atmosphere is controlled for one of two reasons: to prevent surface reaction or to cause a surface reaction, such as carburizing or nitriding.

Prevention or control of oxide layer formation is primarily a matter of controlling the oxygen partial pressure of the atmosphere at a value low enough to prevent oxidation, as described in Section 7.2.1. For a metal that undergoes oxidation according to the reaction shown in Eq. 7.2 (Section 7.3), where MO is the lowest oxide of M, the oxygen partial pressure must be controlled so as not to exceed a value :

7.84

where ΔG° is the standard free energy of reaction 7.2.

Unfortunately, is a function of temperature and has lower values at lower temperatures. Thus, if an atmosphere is designed to be effective at high temperatures, it may become oxidizing as the temperature is reduced during cooling. A surface oxide layer may, therefore, form as the metal is cooled. Although the metallurgical damage to the surface will be negligible, the surface may be discolored, i.e. not bright. This condition can be overcome to some extent by rapid cooling or by changing the atmosphere to a lower oxygen partial pressure just before or during cooling.

For alloys, the most critical reaction must be considered when deciding on the composition of the atmosphere to be used. For this purpose, the activities of the alloy components must be known since, if the metal M in Eq. 7.2 exists at an activity a_M, the corresponding equilibrium oxygen partial pressure will be given by :

7.85

If the metal activities in the alloy are not known, then, by assuming the solution to be ideal, mole fractions may be used instead of activities to give a value of the oxygen partial pressure at which experiments must be performed to establish the correct atmosphere composition.

Low oxygen partial pressures can be provided and, more importantly, controlled by using “redox” gas mixtures. These mixtures consist of an oxidized and a reduced species, which equilibrate with oxygen, e.g.

7.86

from which or, more importantly, can be obtained:

7.87

7.88

Thus, from Eq. 7.87, the ratio of carbon dioxide to carbon monoxide may be calculated for any oxygen partial pressure and temperature.

Further discussion of protective atmospheres can be found in references by Jansson and Foroulis (1973) and Kofstad (1966, 1988).

7.7.2 Alloying

For engineering applications, metals are strengthened and their environmental resistance improved by appropriate alloying. The basic mechanisms operating in pure metal oxidation are also operative in the oxidation of alloys with added complications. These complications include the formation of multiple oxides, mixed oxides, internal oxides, and diffusion interactions within the metals. The effect of alloying on oxidation behavior can be understood first by considering binary and ternary alloy systems, upon which so many commercial high temperature materials are based.

Alloy components have different affinities for oxygen and do not diffuse at the same rate in the oxide or the alloy. Consequently, the simple kinetic rate equations are often not followed, and the scale and the alloy compositions change in a complex way with time. The second component may enter the scale, affecting its lattice defect structure, or may accumulate as metal or as oxide beneath the main scale. Also, if oxygen diffuses into the alloy in atomic form, precipitation of the oxide of the less noble metal may occur as internal oxide. When it is considered that scales can crack, contain voids, spall, sinter, and give multiple layers of irregular thickness, it is clear that the general situation is complex. It is important to attempt to break alloy oxidation down into certain limiting cases, applicable specifically to binary alloys but in a more general way to commercial alloys, which can be treated quantitatively or at least semiquantitatively.

Thermodynamics, usually in the form of ternary equilibrium diagrams involving two alloying components and oxygen (or sulfur, etc.), are useful in predicting the alloying element likely to be preferentially oxidized and sometimes the likely steady‐state scale. They can also provide information on subsequent reactions, such as the changes occurring when internal oxide is incorporated into the main scale. The application of thermodynamic concepts is exemplified here for a binary alloy AB, based on metal A, which is less reactive than the alloying element B. The oxides formed will be referred to as AO_a and BO_b.

7.7.2.1 Basic Thermodynamic Approach

Let AB be an alloy for which, at temperature T, A and B form an ideal solid solution but two immiscible oxides, AO_a and BO_b. Let be the standard Gibbs free energy, at this temperature, of the following reaction:

7.89

For a binary alloy AB in which N_B is the mole fraction of the solute B, the equilibrium oxygen pressure obeys the following relationship:

7.90

The variation of the oxygen pressure of the alloy composition, at a given temperature, is plotted in Figure 7.14 (curve (1)) (Gesmundo and Niu 1998). Likewise, the variation of the oxygen pressure of the following equilibrium ( is the standard Gibbs free energy of the reaction)

7.91

obeys the equation

7.92

The corresponding variation of the oxygen pressure with the alloy composition is also drawn in Figure 7.14 (curve (2)).

The point of intersection between curves (1) and (2) (point E in Figure 7.14) corresponds to equilibrium between both oxides AO_a, BO_b and the alloy (at an equilibrium pressure termed P_t). The mole fraction of component B, corresponding to point E, depends on the standard Gibbs free energy of reactions 7.89 and 7.90, according to

7.93

Figure 7.14 was drawn assuming that the solute B is much more reactive with oxygen than the base component A. In that case, and the mole fraction corresponding to point E is very much less than 0.5.

It is necessary to plot the diagrams of condensed phases to check the compatibility of the proposed models with the experimental results.

In Chapter , many thermodynamic diagrams for high temperature corrosion situations are considered in detail, and here most emphasis is being placed upon factors affecting the spatial distribution of components in the scale and alloy and the influence of these factors on overall scaling behavior.

When an alloy is oxidized, the oxides of the components may be completely miscible, producing an oxide solid solution, or they may be completely or partly immiscible, producing multiphase scales. In the case of partially or completely immiscible oxides, a crude categorization for a range of alloys AB, in which A is the more noble metal and B the less noble metal, is as follows:

1. A relatively narrow composition range near pure A where AO is produced almost exclusively, at least in the external scale.
   
2. A relatively wide composition range near pure B where BO is produced exclusively.
   
3. An intermediate composition range where both AO and BO are produced.

In composition ranges (1) and (3), it is often possible to get doping of the major oxides. For example, if AO is produced almost exclusively, small concentrations of B may dissolve in the oxide, even if B is of a different valency to A and BO of different crystallographic type to AO, thereby changing its defect structure and growth rate. Furthermore, in the intermediate composition range, AO and BO may combine to give partial or complete layers of a more complex oxide compound.

There are two distinct possibilities of binary alloy oxidation. Initially, the special case of the exclusive oxidation of one alloy component is considered before procedure to the more general case where both components oxidize (Figure 7.15).

Group I

Only one of the following elements oxidizes under the prevailing conditions, giving BO.

1. The addition (minor) element B oxidizes under the given conditions:
   
   
   
   1. Internally, giving BO particles in a matrix of A, e.g. Ag–Si alloys dilute in silicon, producing SiO₂ particles in a silver matrix (Figure 7.15a). The oxygen pressure in the atmosphere is less than the equilibrium dissociation pressure of AO.
      
   2. Exclusively externally, giving a single layer of BO above an alloy matrix depleted in B, e.g. Ag–Si alloys richer in silicon, producing an external SiO₂ layer (Figure 7.15b). The oxygen pressure in the atmosphere is again generally less than the dissociation pressure of AO. A special case of this situation exists, however, where both alloying elements can oxidize, but the kinetic and geometrical conditions allow B to be highly selectively oxidized, e.g. Fe–Cr and Ni–Cr alloys reasonably rich in chromium produce essentially Cr₂O₃ scales, especially when oxidized at low partial pressures of oxygen. In practice, small amounts of iron or nickel enter the Cr₂O₃ scales, producing solid solutions of doping effects.
   
   
2. The element B is now the major element and oxidizes exclusively:
   
   
   
   1. Leaving the non‐oxidizable metal A dispersed in BO, e.g. Cu–Au alloys rich in copper (Figure 7.15c).
      
   2. Leaving the non‐oxidizable metal A in an A‐enriched zone beneath the BO scale, e.g. Ni–Pt alloys (Figure 7.15d), Fe–Cr alloys richer in chromium than those in group I (a)(2) come in this category, as do certain Fe–Ni alloys under appropriate conditions.

Group II

Both alloying elements oxidize simultaneously to give AO and BO, the oxygen pressure in the atmosphere being greater than the equilibrium dissociation pressures of both oxides.

1. AO and BO react to give a compound:
   
   
   
   1. AO and BO give a single solid solution (A, B)O, e.g. Ni–CO alloys (Figure 7.15e). In practice, some internal (A, B)O richer in B than the surface scale is found in the alloy.
      
   2. A double oxide is formed, often as a spinel, which may give a complete surface layer of variable composition (Figure 7.15e), as for certain Fe–Cr alloys, or particles incorporated into a matrix of AO if the reaction is incomplete, as for certain Ni–Cr alloys (Figure 7.15f).
   
   
2. AO and BO are virtually insoluble in each other:
   
   
   
   1. The less noble metal B is the minor component. An internal oxide of BO lies beneath a mixed layer of AO and BO, e.g. certain Cu–Ni, Cu–Zn, Cu–Al alloys, and many other examples (Figure 7.15g).
      
   2. The less noble metal is the major component so that no internal oxidation is now observed (Figure 7.15h). In practice, the second‐phase oxidation in Figure 7.15g may not be present in the outer regions of the scale because AO may grow rapidly to produce outer regions exclusively of this oxide. The outer regions may be oxidized to higher oxides, e.g. a CuO layer is found outside Cu₂O on Cu–Si alloys. Conditions may develop so that the internal oxide particles link up to give a complete healing layer of BO at the scale base. This can apply to alloys that begin scaling as in Figure 7.15g.

It is therefore apparent that there is not, and never can be, a single comprehensive theory of alloy oxidation. Rather, there is a sequence of special cases. Furthermore, it must be recognized that in certain alloy systems several classifications of behavior are possible, depending on the composition and oxidation atmosphere, temperature, pressure, time, etc. Changes from one type of behavior to another may occur on a single specimen.

The case of alloys AB, such as Ag–In or Ag–Si, in which only one of the components can be oxidized at elevated temperature, is considered first. The component B on the alloy surface produces nuclei of BO in a matrix of A. If B can diffuse up to the alloy surface sufficiently rapidly, a complete surface layer of BO is produced, but if this condition is not met, atomic oxygen diffuses into the alloy, precipitating BO internal oxide particles at appropriate locations. So, the formation of BO internally or externally depends on the balance between the outward flux of B and the inward flux of oxygen in the alloy.

The process of internal oxidation occurs in the following manner. Oxygen dissolves in the base metal (either at the external surface of the specimen or at the alloy–scale interface if an external scale is present) and diffuses inward through the metal matrix containing previously precipitated oxide particles. The critical activity product, a_Ba_O, for the nucleation of precipitates is established at a reaction front (parallel to the specimen surface) by the inward‐diffusing oxygen and the outward diffusion of solute (reaction B + O = BO). Nucleation of the oxide precipitate occurs, and a given precipitate grows until the reaction front moves forward and depletes the supply of solute B arriving at the precipitate. Subsequent precipitate growth occurs only by capillarity‐driven coarsening (Ostwald ripening).

In summary, the main conditions that favor internal oxidation are as follows:

1. The alloying element is less noble than the base alloy.
   
2. Oxygen permeability within the base alloy is sufficiently high. Note that the measurement of the depth, δ (of the subsurface zone where particles of the oxide BO are dispersed within the metal matrix A), for an alloy with a known concentration of element B allows us to estimate the product , which is the oxygen permeability within the alloy. Here, is the oxygen concentration (moles/unit volume) at the alloy surface in equilibrium with the gaseous atmosphere ( at t = 0), and D_O is the oxygen diffusion coefficient.
   
3. Oxygen dissolution within the alloy is not inhibited by a surface layer of oxide.
   
4. The mole fraction of alloying element B is sufficiently low.

In this model of internal oxidation and under steady‐state conditions, with a stable BO oxide formed, the number of moles of element B that react under unit area of the specimen surface is equal to the number of moles of diffusing oxygen:

7.94

where is the value of C_B for x ≥ 0 (at t = 0) and x → ∞ (for whatever t). By integration, a parabolic law for the depth, δ, of oxidation is obtained:

7.95

which shows that the depth of oxidation zone at a given time is inversely proportional to the square root of the concentration of element B in the alloy. The penetration rate of the oxidation zone decreases with increasing concentration of element B and decreasing oxygen concentration, , at the alloy surface produced, for example, by decrease in the oxygen partial pressure of the oxidizing gas. It can be appreciated that there is a limiting concentration of element B above which a continuous blocking oxide scale is formed. This corresponds to the transition from internal to external oxidation and a consequent decrease in the oxidation rate. The main conditions that favor external oxidation are as follows:

1. (i) The mole fraction of alloying component B is sufficiently high, and/or the diffusion coefficient D_B is high.
   
2. (ii) Oxygen permeability within the base alloy is low.

Wagner has proposed a criterion based on the comparison of both the diffusion coefficients and the equilibrium concentrations of B and O in the alloy. For an oxide BO the criterion is expressed, in a simplified form, as the ratio between the depths of the B‐depleted zone (external oxidation) and the internally oxidized zone:

7.96

γ > 1 leads to external oxidation, whereas internal oxidation occurs when γ < 1.

The transition is the basis for the design of alloys based on iron, nickel, and cobalt and containing elements such as Cr, Al, and Si, which give highly stable oxides, e.g. Cr₂O₃, Al₂O₃, and SiO₂. With a sufficiently high content of alloying component, a complete external scale of oxide is formed, which protects the alloy from further oxidation. It should be pointed out that with alloying concentrations around the critical value, mechanical damage of the protective scale may result in transient non‐protective oxidation of the solute‐depleted zone.

The process described above in which a solute oxidizes preferentially to the parent element and forms a continuous layer on the surface is referred to as selective oxidation. The selective oxidation of elements that form a slowly growing protective layer is the basis for the oxidation protection of all alloys and coatings used at high temperature. The only elements that consistently result in protective scales are Cr (chromia scale), Al (alumina scale), and Si (silica scale). However, SiO₂ is not stable at low pressures. It decomposes to gaseous species such as SiO. It also reacts with water vapor at high temperatures, forming Si(OH)₄ gas. The use of Cr₂O₃ scale‐forming alloys as well as Al₂O₃ scale‐forming alloys requires a further analysis.

7.7.2.1.1 Chromia‐Forming Alloys

There are three composition (in weight %) regimes, with distinct oxidation characteristics for each family of Ni–Cr forming a chromia scale:

• Ni–Cr < 10%: Such alloys form a NiO external scale and internal Cr₂O₃.
  
• Ni‐30% > Cr > 10%: In such alloys the outer scale consists of NiO on grains and Cr₂O₃ in grain boundaries.
  
• Ni–Cr > 30%: For such alloys the external scale consists of Cr₂O₃, which is maintained because of the large Cr reservoir.

The oxidation of Fe–Cr and Co–Cr alloys is similar to the Ni–Cr oxidation, leading to chromia scales, apart from Fe₂O₃, Fe₃O₄, and other oxides (Gesmundo and Viani 1986; Wood and Chattopadhyay 1970; Wood and Stott 1987; Wood et al. 1970).

7.7.2.1.2 Alumina‐Forming Alloys

There are also three composition regimes, with distinct oxidation characteristics for each family of Ni–Cr forming an alumina scale:

• Ni–Al < 6%: Such alloys form a NiO external scale and internal Al₂O₃ and NiAl₂O₄.
  
• Ni‐17% > Al > 6%: In such alloys the outer scale initially consists of Al₂O₃. However, on continuous exposure, Al depletion occurs in the alloy adjacent to the oxide scale. In the depleted zone, the Al activity is well below the requirement to form continuous Al₂O₃. NiO, therefore, overtakes Al₂O₃. The overall result is the formation of a mixture of NiO and NiAl₂O₄ spinel formed by the combination of NiO and Al₂O₃.
  
• Ni–Al > 17%: For such alloys, the external scale of Al₂O₃ is maintained because of the large Al reservoir.

The oxidation of Fe–Al, Fe–Cr–Al, Ni–Cr–Al–Y, Co–Cr–Al–Y, and Fe–Cr–Al–Y alloys is similar to the Ni–Al oxidation, leading to alumina scales, apart from other Fe₂O₃, Fe₃O₄, Y₂O₃, and other oxides (Pint and Hobbs 1994; Pint et al. 1997; Westbrook and Fleischer 1994).

Alumina, Al₂O₃, exhibits a very low deviation from stoichiometry, much lower than chromia, Cr₂O₃, therefore ensuring more protective behavior (Table 7.5).

	Al2O3	Cr2O3
Deviation from stoichiometry	10−4 (at 1000 °C)	10−3 (at 600 °C)
Cation self‐diffusion coefficient in oxide at 1000 °C (cm2 s−1)	10−15	10−12
Parabolic rate constant at 1000 °C (on Fe–25Cr–5Al for Al2O3, on Fe–20Cr for Cr2O3) (g2 cm−4 s−1)	2.7 × 10−13	2.0 × 10−11

The addition of Cr to Ni–Al alloys results in a remarkable synergistic effect that is of great technological interest. For example, chromium additions of about 10 wt% can enable external Al₂O₃ formation on alloys having aluminum levels as low as 5 wt%. This phenomenon has allowed the design of more ductile alloys and coatings.

The oxidation resistance of high temperature materials is so dependent on the formation of a chromia or alumina layer that it seems appropriate to further discuss the characteristics of these protective oxides.

7.7.2.1.3 Chromia and Alumina Characteristics

Cr₂O₃ has the corundum crystal structure (Figure 7.16) consisting of a hexagonal lattice, close packed with oxygen anions with 2/3 of the octahedral holes occupied by Cr. It is the only relevant oxide stable at high temperatures and is protective against both corrosion and oxidation at elevated temperatures. The Cr(III) oxide is an intrinsic semiconductor above 1250 °C that, depending on the temperature, can be p‐ or n‐type.

Lillerud and Kofstad (1980) showed that the oxide has an equal concentration of holes and electrons at temperatures above 1000 °C, depending on the oxygen partial pressure, whereas at lower temperature chromia is p‐type or n‐type (at low partial pressures). Cr₂O₃ is cation deficient, and it increases at a(O₂)^1/8 rate (Stott 1987). Cr interstitials are the predominant ionic defect at low (Young et al. 1985), whereas Cr vacancies are predominant (Stott 1987). Certainly, most studies of chromium oxidation indicate that the scale grows largely by chromium diffusion outward, as confirmed by oxygen tracer studies in the growing scale. However, oxygen isotrope exchange measurements (Lees and Calvert 1976) have inferred that anion transport by mechanisms other than lattice diffusion does not occur during growth of Cr₂O₃ scales on alloys at high temperature. Rapid‐diffusion paths, such as grain boundaries, have a significant influence on the oxidation behavior of chromium at high temperature, as they enhance the transport of cations and perhaps anions (Caplan and Sproule 1975). The scale microstructure is also determined for the chromia oxidation rate, as the fine‐grained structure increases the number of short diffusion paths, therefore allowing higher oxidation rate to be achieved. The addition of reactive elements, such as yttrium, hafnium, cerium, zirconium, etc., improves the oxidation behavior as well as scale adhesion of Cr₂O₃ (Ecer and Meier 1979) and their oxides, present as fine dispersed particles in the alloy prior to oxidation (Hindam and Whittle 1982). At very high temperatures, chromia reacts with oxygen to form the volatile species CrO₃, which induces oxide thinning and a metal recession, when the diffusion through the scale is low. This effect becomes significant at 1000 °C. Therefore, alumina is a suitable candidate, as there is no vapor in the Al–O₂ system with significant vapor pressures.

Aluminum oxide exists in various modifications, depending on the temperature and the exposure time. These phases (γ‐Al₂O₃, θ‐Al₂O₃, and δ‐Al₂O₃) are mostly metastable and are called transition oxides, as they are all transformed to the stable α‐alumina after prolonged exposure. α‐Al₂O₃ is also known as corundum, as its crystal structure is corundum, such as chromia (Figure 7.16). In this case, the Cr ions are replaced by Al cations located in octahedrally coordinated interstitial sites in the anion sublattice and fill 2/3 of the interstices in order to maintain electrical neutrality.

The resistance of materials at high temperature is defined by the diffusion of the species through the oxide as the kinetics depends on the mobility of the ions. It has been shown that alumina grows from both outward diffusion of Al³⁺ and inward diffusion of oxygen with a similar diffusion rate. The protection given by α‐Al₂O₃ scales at high temperatures is due to the low concentrations and mobility of ionic and electronic defects. The concentrations of intrinsic point defects are probably so low that all reported properties relating to point defects and transport have been dominated by impurity dopants. It is believed that ionic defects are the major species, with electrons and holes as minorities. The concentrations of all defects depend on doping and oxygen pressure, but the nature of the dominant native ionic defects is not well established (Kröger 1981). The diffusion of aluminum occurs through bulk lattice and oxide at the grain boundaries, which represents the major short‐circuit path (Reddy et al. 1982), whereas the oxygen diffusion in the scale is less prominent. The scale formed has a dual morphology, whereby the outer region usually consists of small and equiaxed grains, while the grains are columnar and coarser in the inner region (Felton and Pettit 1976; Smialek 1978). The scale morphology changes with temperature: at 900 °C whiskers are observed on the grown oxide at the oxide–gas interface extending from the substrate surface, which may be due to the formation of the unstable θ‐Al₂O₃, whereas an almost flat layer (α‐Al₂O₃) is observed at 1200 °C (Prescott and Graham 1992). The scale growth mechanisms depend strongly on the exposure temperature.

One of the major concerns for protective oxides at high temperatures is their adhesion to the alloy. The adhesion is mostly lost due to stresses induced during scale growth. Generally, the thermal expansion between metal and oxide is large enough to cause compressive stresses (Hindam and Whittle 1983) at room temperature, causing an interfacial shear stress and subsequent scale buckling (Hindam and Whittle 1982; Sadique et al. 2000). In the case of chromia, the addition of reactive elements (≤0.01%) improves the oxidation behavior and scale adhesion of alumina (Anderson et al. 1985). The effects of rare earth elements on the oxidation resistance at high temperature are various. Reactive elements react with sulfur and form stable sulfide (Smeggil 1987), and sulfur tends to weaken the surface (Sigler 1988). They also promote faster formation of alumina.

From the above considerations, it is clear that native and doped ionic defects in compact and multilayered scales represent important factors controlling the oxidation rate of binary and multicomponent alloys at high temperature. In particular, there is now evidence that doping is the most likely rate‐controlling factor in many complex oxidation processes. Therefore, a short reference is now given to doping and the Wagner–Hauffe rules.

7.7.2.1.4 Doping on Oxide Defect Structure

The various types of defects in oxides have been well described by Kofstad (1966) and neglecting rare oxides; it can be said that the commonly found nonstoichiometric, semiconducting oxides are normally divided into two general classes:

1. Oxides with cation defects:
   
   
   
   1. Metal deficient, with cation vacancies on the cation sublattice (p‐type semiconductor), e.g. NiO, CoO, and FeO.
      
   2. Metal excess, with interstitial cations (n‐type semiconductor), e.g. possibly ZnO.
   
   
2. Oxides with anion defects:
   
   
   
   1. Oxygen deficient, with oxygen ion vacancies on the anion sublattice (n‐type semiconductor), e.g. Nb₂O₅ or Ta₂O₅.
      
   2. Oxygen excess, with interstitial oxygen ions (p‐type semiconductor), e.g. UO₂.

In actuality, the defect structures are often more complex, the oxides containing several types of defects, sometimes linked with impurities.

When a foreign on or dopant is added to a nonstoichiometric oxide, electroneutrality must be maintained by a redistribution of electronic and ionic defects. For example, with a metal‐deficient p‐type oxide like NiO, addition of trivalent ions such as Cr³⁺ increases the concentration or activity of cation vacancies but decreases the concentration of electron holes. Since transport of cations through the cation vacancies is rate determining in scale growth, there is an observed increase in oxidation rate. Additions of monovalent ions such as Li⁺ have the reverse effect and reduce the oxidation rate. Comparable rules would apply to oxygen‐excess p‐type semiconductors. For n‐type oxides of either type, additions of cations of a higher valency than those of the parent oxide decrease the respective concentrations or activities of anion vacancies or cation interstitials, thereby generally reducing the oxidation rate. Lower valency additions increase the oxidation rate. Addition of ions of the same valency should have little effect at such low levels.

The Wagner–Hauffe rules should, in principle, apply to individual matrix and precipitate phases in multiphase oxides as well as to single‐phase oxides, as long as blocking effects of precipitates and certain other general rules regarding the nature of the solution of the dissolving ions are obeyed. The overoptimistic application of the rules has led to disappointment in their lack of generality, but closer scrutiny often shows that obedience could not reasonably be expected in real oxidation situations. For the rules to be obeyed, the true defect structure of the parent oxide must be fully understood (many oxides really contain several defects, possibly in association), the foreign cations are assumed to enter normal cation positions in the parent oxide at their anticipated valency, and a Wagner‐type parabolic growth relationship must hold rather than a mechanism involving, say, grain boundary diffusion. Ions often have only limited solubility in the parent oxide, so the rules would not necessarily hold progressively at high addition levels. The ionic radius is a pertinent parameter in such considerations. In ideal cases, it is assumed that the dopant is homogeneously distributed through the scale, i.e. the parent and dopant oxidize at the same rate, and the rates of diffusion of the two cations in the scale are equal. In practice this probably rarely, if ever, occurs, and, except in special circumstances, concentrations of ions at the alloy–oxide and oxide–oxygen interfaces may change with time. Recently several cases of obedience and disobedience to the rules are being summarized.

Recent examples that were once considered classical cases of such doping, including the increase in oxidation rate of nickel and cobalt by additions of chromium, are now known to be much more complex. The increase in oxidation rate is a result of a very soluble interplay of doping, or other alloying effects in the semiconducting oxide, possibly electric field effects, internal and external oxidation, effects on vacancy consumption by the alloy and hence the activity gradient across the scale, pore formation in the inner and possibly the outer regions of the scale, and finally the reduction of cross‐sectional area within the scale and at the alloy–oxide interface caused by Cr₂O₃ or NiCr₂O₄ or CoCr₂O₄ spinel particles (Kofstad and Hed 1969; Wood and Hodgkiess 1966).

Commercial superalloys contain many alloying elements of significance over and above chromium and aluminum. The oxidation behavior of these alloys is very complex, and oxidation resistance varies widely, although the general mechanisms described earlier still apply. The complexities arise from significant influence of the individual elemental constituents. Nickel‐based superalloys containing elements such as Co, Cr, Al, Ti, W, and Ta exhibit general behavior similar to simple NiCrAl alloys. These superalloys are the most common materials used for high temperature applications. NiCrAlY‐type superalloy coatings also exhibit two interesting properties: they can be adjusted to have compositions near that of the substrate, avoiding differential strain problems and possible loss of adhesion, and they provide high resistance to oxidation due to the combined effects of Al and Y. This subsection on alloying finishes by reporting basic aspects of nickel‐based superalloys, taking into account their increasing importance as industrial alloys for high temperature applications.

7.7.2.1.5 Ni‐Based Superalloys

Superalloys belong to the category of high temperature materials with excellent mechanical properties and oxidation resistance at room and elevated temperatures. They also possess a structural stability at high temperature. In 1973, a superalloy was defined as “an alloy developed for elevated temperature service, usually based on group VIIIA elements, where relatively severe mechanical stressing is encountered, and where high surface stability is frequently required” (Sims and Hagel 1972). This definition is still applicable. Superalloys are widely used in aerospace, power generation, aircraft, marine, petrochemical processing, gas and oil extraction as rocket engines, nuclear reactors, and steam power plants, just to name a few of their applications.

The main superalloys are Fe‐, Ni‐, and Co‐based alloys. At temperatures up to around 700 °C, ferritic Fe‐based superalloys are used. With the highest strength, Co‐based superalloys can be used at very high temperatures but have poor corrosion resistance at high temperatures.

Ni‐based superalloys can be applied at higher temperatures, whereas single crystalline materials can be used at 0.8T_m (slightly below the melting point) and for durations of up to 100 000 hours (Sims et al. 1987) and even for 200 000 hours for special alloys (Reed 2006) at somewhat lower temperatures. These materials are generally used in the hottest areas, such as the hottest part of turbines, due to their very high strength. Their metallurgy is very complex as they can be fabricated as either wrought alloys, cast alloys, or directionally solidified alloys. The chemical composition of Ni‐based superalloys is very complex as up to 12 elements can be alloyed to Ni, owing to its high alloying tolerance. Furthermore, the “tramp” elements – such as sulfur, phosphorus, oxygen, and nitrogen – can be carefully controlled through appropriate melting practice. Trace elements, such as selenium, bismuth, thallium, tellurium, and lead, should be held at very low levels. With the addition of chromium and aluminum, these alloys can form continuous and adherent scales – alumina at very high temperatures and chromia at somewhat lower temperatures. Cr₂O₃ is also very effective against hot corrosion. The former polycrystalline alloys have very high Cr content (10–20% Cr), which has been decreased with the increasing use of alloys at higher temperatures. In contrast, Al content has been maintained at around 6%. The quantity of elements, such as molybdenum, tungsten, tantalum, columbium, and hafnium, depends on the strength that might be reached. With the advent of single crystalline materials, the grain boundary strengtheners – boron, zirconium, and carbon – have been removed, or their content has been reduced, carbon in particular, as is the case in, for example, the single crystalline superalloy PWA 1483. Boron and zirconium are known to retard the generation of boundary cracking, as they segregate at the grain boundaries. Zr is also a “getter” center for trace and tramp elements. Hafnium has taken the role of Zr in the new generation of alloys.

Superalloys have a close‐packed FCC (face‐centered cubic) austenite lattice. The lattice consists generally of a γ′‐phase (strengthening phase Ni₃Al or Ni₃Ti with Al or Ti positioned at the corners and Ni on the faces), which are represented in Figure 7.17.

Other phases such as γ″, carbides, borides, TCP (topologically closed packed phases) should also be mentioned. The alloying elements are selectively distributed in the crystal.

The γ‐matrix usually contains a high percentage of solid solution elements, such as Co, Cr, Mo, W, Ta, Ti, and Al, strengthening the matrix.

The γ′ phase usually consists of (Ni, Co)₃(Al, Ti) with a higher amount of Al or Ti. The superlattice of Ni₃Al is the Cu₃Au (L1₂)‐type structure. This phase precipitates coherently with gamma. This intermetallic phase has the particular characteristic of increasing its creep with increasing temperature, whereby its ductility prevents cracks. A small misfit between γ/γ′ is of importance, as it lowers the total surface energy, therefore stabilizing the microstructure at high temperatures.

Carbides are more important for polycrystalline materials (low content 0.05–0.2% carbon) as they precipitate at grain boundaries and react with refractory elements (Ti, Ta, and Hf) to form strong carbides (MC, M is the metal), which strengthen the grain boundary, prevent or retard grain boundary sliding, or allow stress relaxation. They may also tightly bind some elements that favor phase instability.

Borides may also have a similar role as carbides. Boron (50–500 ppm) is located at grain boundaries and provides enhancement of the grain boundary cohesion, thus reducing grain boundary tearing under creep‐rupture loading.

TCP phases, which are platelike phases, including σ, μ, and Laves forms, result from excessive alloying of elements, such as Mo, W, Re, and Cr. They are undesirable in Ni‐based superalloys as they detrimentally affect the properties of the alloys. Due to their morphology, they are an excellent source for crack initiation and propagation, leading to low temperature brittle failure, especially sigma, which contains a high amount of refractory metals sapped from the γ‐matrix, causing the loss of solution strengthening.

The development of new metallurgical processes (directional solidification and single‐crystal formation) has improved the performances of gas turbines. Single crystals, which are widely used in the hottest part of turbine (e.g. in aircraft turbines), are based on the γ/γ′ constituent. The volume fraction of γ′ is generally in the range of 40–70%. This phase is either distributed in spherical shape, if the lattice mismatch is around 0–0.2%, or cubical, occurring at 0.5–1.0% lattice mismatch (Sims and Hagel 1972), or plates become perpendicular to the load at mismatch about 1.25% (Fedelich 1999). Needles aligned with the load, as well as plates, have also been observed by Bressers (1996) as a consequence of anisotropic coalescence due to extreme temperatures and loading conditions.

Some elements, such as the grain boundary strengtheners – boron, zirconium, and carbon – are removed in single crystalline materials, as these do not have grain boundaries. Both Zr and Hf tend to lower the solidus temperature of the alloy, preventing the complete dissolution of γ′ during solution heat treatment. Hf is, however, added in very low quantities (0.01–0.05%), as it reacts with sulfur (the presence of which is undesirable) to form stable sulfides, therefore improving the scale adhesion of the oxides. In addition, it prevents cracking during the solidification cool‐down cycle in cored columnar‐grained alloys.

After service, some elements accumulate at the dendritic core or in interdendritic regions during cooling. Al, Ti, and Ta tend to segregate in the interdendritic parts, whereas Re and W, for example, join the dendritic core (Fuchs 2001, 2002), which can be reduced by an appropriate solution treatment. The amount of refractory metals in superalloys, such as W, Ta, and Re, has been steadily increasing in the past few years, as they increase the creep resistance as well as strength due to solid solution strengthening, strengthening of γ′, and slower diffusion rate. The new generation of superalloys also contains rhenium, which not only increases the mechanical properties of the alloy (Pollock 2000) but also leads to the formation of TCP phases (Feng et al. 2003).

Superalloys, just as most of high temperature alloys, owe their good oxidation resistance to the formation of a protective scale, which is slow growing, has a low defect concentration, and is adherent. This is achieved by the selective oxidation of either Cr or Al, depending on the service temperature. Therefore, a minimum amount of these elements is required to ensure the growth of this external scale. Due to its volatility at around 1000 °C, Cr₂O₃ is replaced by α‐Al₂O₃, which is stable from about 950 °C. On the other hand, the presence of Cr decreases the amount of Al needed to form alumina rapidly (Prescott and Graham 1992). The oxides formed separate the substrate with the corrosive medium, therefore preventing the formation of fast‐growing oxides, such as Ni, Co, or Fe oxides. A better adherence of the scales is mostly achieved by addition of reactive elements, such as hafnium, yttrium, and zirconium, as alumina tends to be very brittle. The selective oxidation of Cr and Al lead to their depletion under the alloys, causing the degradation of the mechanical properties especially if the substrate is very thin.

7.7.3 Protection by Coatings

The focus on the development of substrate alloys discussed in this chapter is generally to achieve high strength, high ductility, and efficient production. Oxidation resistance may not be consistent with achieving these goals. For example, increased Al and Cr result in improved oxidation resistance; however, beyond a certain level, these elements reduce creep strength of the resulting alloys. To achieve both strength and resistance to environmental degradation, the two functions are separated. The load capability is provided by the application of thin coatings with adequate Al and Cr. The thickness of the coating is controlled so that it does not carry any significant load. Depending on the temperature of use, many high temperature alloys require coatings compatible with its composition and structural (modulus) and thermal (CTE) properties. Diffusion, overlay, and thermal barrier coatings were described a few years ago in a chapter on high temperature oxidation that we have published in Uhlig’s Corrosion Handbook, 3rd ed., edited by R. Winston Revie. Recently, surface coatings are gaining high potential industrial applications in the field of high temperature corrosion protection, Therefore, it was felt that protective coatings should be discussed in more detail deserving a chapter on the subject, as it is done in Chapter of this book. Here, surface coatings are considered in terms of coating systems, coating processes, and coating degradation and are extended for cyclic oxidation, hot corrosion, sulfidation, carburization, erosion–corrosion, etc.

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Further Reading

1. Aaronson, H.I. (ed.) (1999). Lectures on the Theory of Phase Transformations. Warrendale, PA: The Minerals, Metals and Materials Society.
   
2. Arper, A.M. (ed.) (1970). Phase Diagrams, Materials Science and Technology, vol. 1. New York: Academic Press.
   
3. Ashby, M. and Jones, D.R.H. (2005). Engineering Materials 1: An Introduction to Properties, Applications and Design, 3e. London: Elsevier.
   
4. Baboian, H. (ed.) (1995). High‐Temperature Gases (Corrosion Tests and Standards: Applications and Integration). Washington, DC: ASTM.
   
5. Bak, T., Nowotny, J., and Sorell, C.C. (1997). Key Eng. Mater. 125–126: 1.
   
6. Barralis, J. and Maeder, G. (1997). Précis de Métallurgie: élaboration, structures‐proprietés, normalisation. Paris: AFNOR‐Nathan.
   
7. Belton, G.R. and Worrell, W.L. (1970). Heterogeneous Kinetics at Elevated Temperatures. New York: Plenum Press.
   
8. Böhm, G. and Kahlweit, M. (1964). Acta Metall. 12: 641.
   
9. Bose, S. (2007). High Temperature Coatings. Amsterdam, Holland: Elsevier.
   
10. Dahotre, N.B. and Hampikian, J. (eds.) (1999). Elevated Temperature Coatings: Science and Technology III. San Diego, CA: TMS.
    
11. Darken, L.S. and Gurry, R.W. (1953). Physical Chemistry of Metals. New York: McGraw‐Hill.
    
12. Donachie, M.J. Jr. and Donachie, S.J. (2002). Superalloys: A Technical Guide, 2e. Materials Park, OH: ASM International.
    
13. Douglass, D.L. (ed.) (1971). Oxidation of Metals and Alloys. Materials Park, OH: ASM.
    
14. Douglass, D.L. (1995). Oxid. Met. 44: 81.
    
15. Ellingham, H.J.T. (1944). J. Soc. Chem. Ind. 63: 125.
    
16. Gao, W. (ed.) (2008). Developments in High Temperature Corrosion and Protection of Materials. Cambridge: Woodhead Publ. Ltd.
    
17. Glaser, B., Rahts, K., Schorr, M., and Schütze, M. (1994). Sonderband des Praktischen Metallographie, vol. 25, 75. Frankfurt, M.: MAT‐INFO, Werkstoff‐Informationsges.
    
18. Hauffe, K. and Ilschner, B. (1954). Z. Elektrochem. 58: 382.
    
19. Hauffe, K. (1957). Oxydation von Metallen und Metallegiesungen. Berlin: Springer‐Verlag.
    
20. Herman, H. (ed.) (1971). Advances in Materials Research. New York: Wiley.
    
21. Hocking, M.G., Vasantasree, V., and Sidky, P.S. (1989). Metallic and Ceramic Coatings: Production, High Temperature Properties and Applications. Essex: Longman Scientific & Technical.
    
22. JANAF Thermochemical Tables (1975). J. Phys. Chem. Ref. Data 4: 1.
    
23. Khanna, A.S., Quadakkers, W.S., and Wasserfuhr, C. (1988). The influence of sulfur and its interaction with yttrium on the composition, growth and adherence of oxide scales on alumina‐forming alloys. Conference on the Role of Active Elements in the Oxidation Behaviour of High‐Temperature Metals and Alloys.
    
24. Kofstad, P. (1968). Corrosion 24: 379.
    
25. Landolt, D. (1993). Traité des Matériaux, N° 12, Corrosion et Chimie des Surfaces des Métaux. Presses Polytechniques Romandes.
    
26. Lang, E. (ed.) (1989). The Role of Active Elements in the Oxidation Behaviour of High Temperature Metals and Alloys. Amsterdam: Elsevier.
    
27. Marcus, P. and Mansfeld, F. (eds.) (2006). Analytical Methods in Corrosion Science and Engineering. Boca Raton, FL: CRC Press, Taylor & Francis Group.
    
28. Mitchell, D.F., Hussey, R.J., and Graham, M.J. (1983). J. Vac. Sci. Technol., A 1: 1006.
    
29. Mrowec, S. and Werber, T. (1978). Gas Corrosion of Metals. Varsovie, Poland: Foreign Scientific Publications Department of the National Center for Scientific, Technical and Economic Information.
    
30. Nicholls, J. (ed.) (1999). High Temperature Surface Engineering. London: Institute of Materials.
    
31. Nowotny, J. (1992). Diffusion in Solids and High Temperature Oxidation of Metals. Zürich: Trans. Tech. Publications Ltd.
    
32. Nowotny, J. and Sorrell, C.C. (1997). Electrical Properties of Oxide Materials. Zürich: Trans. Tech. Publications Ltd.
    
33. Psaras, P.A. and Langford, H.D. (eds.) (1987). Advancing Materials Research. Washington, DC: National Academy Press.
    
34. Rahmel, A. (ed.) (1982). Aufbau von Oxidschichten auf Hochtemperaturwerkstoffen und ihre technische Bedeutung. Oberursel: Oberursel, Deutsche Gesellschaft für Metallkunde.
    
35. Rapp, R.A. (1965). Corrosion 21: 382.
    
36. Rickert, H. (1960). Z. Phys. Chem. NF21: 432.
    
37. Sato, Y., Onay, B., and Maruyama, T. (eds.) (1992). High Temperature Corrosion of Advanced Materials and Protective Coatings. Amsterdam, Holland: Elsevier.
    
38. Schick, H. (1966). Thermodynamics of Certain Refractory Compounds. New York: Academic Press.
    
39. Schütze, M. and Quadakkers, W.J. (eds.) (1999). Cyclic Oxidation of High Temperature Materials, EFC27. London: Maney Publishing.
    
40. Schütze, M. and Quadakkers, W.J. (eds.) (2008). Novel Approaches to Improving High Temperature Corrosion Resistance. Cambridge: Woodhead Publ. Ltd.
    
41. Sequeira, C.A.C., Chen, Y., Santos, D.M.F., and Song, X. (2008). Corros. Prot. Mater. 27: 114.
    
42. Stern, K.H. (1996). Metallurgical and Ceramic Protective Coatings. London: Chapman and Hall.
    
43. Streiff, R., Stringer, J., Krutenat, R.C., and Caillet, M. (eds.) (1987). High Temperature Corrosion of Materials and Coatings for Energy Systems and Turboengines. Sequoia, Lausanne, Switzerland: Elsevier.
    
44. Streiff, R., Stringer, J., Krutenat, R.C., and Caillet, M. (eds.) (1993). High Temperature Corrosion and Protection of Materials, vol. 3. Les Ulis: Les Éditions de Physique.
    
45. Streiff, R., Wright, I.G., Krutenat, R.C. et al. (eds.) (2001). High Temperature Corrosion and Protection of Materials, vol. 5. Zürich: Trans. Tech. Publications Ltd.
    
46. Swalin, R.A. (1972). Thermodynamics of Solids. London: Wiley.
    
47. Wagner, C. (1956a). J. Electrochem. Soc. 99: 369.
    
48. Wagner, C. (1956b). J. Electrochem. Soc. 103: 571.
    
49. Wagner, C. (1959). Z. Elektrochem. Soc. 63: 772.
    
50. Wood, G.C. (1978). Oxid. Met. 2: 11.