The Point Defect Model

The electron current density, I, which is sensed in an external circuit, is given by  

Using the method of partial charges, the rate constants for the reactions are found to be of the form [7]  

It caji be assumed that the applied potential changes sinusoidally around some mean value (V) in accordance with Equation (19.7)  

The task, then, is to calculate the faradic admittance, Yp-, which is defined as  

The values of U and other steady-state values can be easily calculated. Assuming some arbitrary value of U, it is possible to immediately calculate i = 4, 5, 7 from Equations (19.5) and (19.6). From the rate equation for the change in thickness of the barrier layer, which is written as 

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Even if the surface is not perfectly smooth, the initial event that must occur in the development of a nucleus is passivity breakdown, in which the protective oxide layer is ruptured to expose the underlying metal to the aqueous environment. The most highly developed theory for this process is the point defect model (PDM) [59-65]. This model postulates that the generation of cation vacancies at the film/solution interface, and their subsequent transport across the barrier layer of the passive film, is the fundamental process fiiat leads to passivity breakdown. Once a vacancy arrives at the metal/film interface, it may be annihilated by reaction (i) in Fig. 31 ... 

Fig. 31, Schematic of physicochemical processes that cwcur within a passive film according to the point defect model m = metal atom Mm = metal cation in cation site Oo = oxygen ion in anion site VjjJ = cation vacancy Vq = anion vaccancy Vm = vacancy in metal phase. During film growth, cation vacancies are produced at the film/solution interface, but are consumed at the metal/film interface. Likewise, anion vacancies are formed at the metal/film interface, but are consumed at the film/solution interface. Consequently, the fluxes of cation vacancies and anion vacancies are in the directions indicated. Note that reactions (i), (iii), and (iv) are lattice-conservative processes, whereas reactions (ii) and (v) are not. Reproduced from J. Electrochem, Sec. 139, 3434 (1992) by permission of the Electrochemical Society.

Fig. 33. Cartoon outlining various stages of pit nucleation according to the point defect model. Reproduced from J. Electrochem, Sec. 139, 3434 (1992) by permission of the Electrochemical Society.

These results show that changes in the anion to cation ratios of the building components of the linked octahedra and tetrahedra which form the core of these silicates is accommodated structurally. There is no evidence to suggest that substantial stoicheiometric variation is accomplished by point-defect populations, and indeed, mineralogists have never had recourse to the point-defect model to account for such changes in stoicheiometry. The brief account above could be greatly expanded and many more examples will be found in the review article previously cited. ... 

In these expressions, by and br are the forward and reverse Tafel constants, respectively, for the metal dissolution reaction, with values of 0.06 V being assumed for both. Actually, they are empirical constants that were assumed a priori in fitting Eq. (9) to the current/voltage data. It is important to note that Eq. (9) applies strictly to Type 304 SS in near neutral solutions [35] and hence that this expression may not be a good empirical model for stainless steels in PWR primary circuits. More recently, the point defect model (PDM) [37] has been used as the basis for... 

Due to low dopant concentrations and high temperatures, the reactions of the doped perovskites with gases may be adequately described by the point defect model and quasi-chemical equilibria ... 

The point defect model of passivity and its breakdown is a variant of the penetration mechanism [51]. The transport of cations from the metal surface to the oxide-electrolyte interface corresponds to an inward movement of cation vacancies Vm+- This inward transport of Vm+ is supported by their high concentration at... 

D.D. Macdonald, The history of the point defect model for the passive state a brief review of film growth aspects, Electrochim. Acta 56 (2011) 1761—1772. 

D.D. Macdonald, Some personal adventures in passivity—a review of the point defect model for film growth, Russ. J. Electrochem. 48 (2012) 235—258. 

D.D. Macdonald, G.R. Engelhard, The point defect model for bi-layer passive films, ECS Trans. 28 (2010) 123-144. 

According to the point defect model, and under conditions where the various equations can be linearized with respect to the applied ac voltage (Vac), the concentration of vacancies at the metal-film and flhn-solution interfaces may be expressed as (Chao et al. [1982])... 

The form of the equation for (Tq is particularly interesting, because it suggests that if the electric field strength (e) and cc are constants (as assumed in the point defect model), then the product (To/dc should be independent of the applied voltage ata oss the system and the thickness of the film. 

While these six generalizations are not all encompassing, in that exceptions may exist, they are sufQcient to differentiate between various theories that have been proposed for the growth of barrier oxide layers on metals and alloys. A number of models that have been developed to describe the growth of anodic oxide films on metals are listed in Table 4.4.2, together with some of their important features and predictions. Of the models listed, which were chosen because they make analytical predictions that can be tested and because they introduced new concepts into the theory of passivity, only the point defect model (PDM) in its latest form (D. Macdonald [1999], Pensado-Rodriguez et al. [1999a,b]) accounts for all of the observations summarized above. 

Figure 4.4.29. Interfacial defect generation/annihilation reactions that occur in the growth of anodic barrier oxide films according to the Point Defect Model (D. Macdonald [1999]). m = metal atom, = cation vacancy on the metal sublattice of the barrier layer, MP = interstitial cation, Mu = metal cation on the metal sublattice of the barrier layer, Vo = oxygen vacancy on the oxygen sublattice of the barrier layer, Oo = oxygen anion on the oxygen sublattice of the barrier layer, = metal cation in solution.

Annihilation Reactions Employed in the Point Defect Model... 

According to the point defect model (D. Macdonald [1999]), the steady state thickness of the barrier oxide layer is given by Eq. (109), which is reproduced here as... 

Table 4.4.6. Parameter Values from Optimization of the Point Defect Model on the Experimental Impedance Data for Alloy-22 in Deaerated, Saturated NaCl (6.2m, pH = 3) Solution at 80°C as a function of applied potential (D. Macdonald et al. [2004]) the parameters apply to the reduced reaction set shown in Figure 4.4.32...

In this chapter, the point defect model (PDM), describing the formation and breakdown of passive films, is reviewed and developed. It is shown how important model parameters can be extracted from experimental impedance data and used to calculate the steady-state barrier layer thickness and passive current density as a function of voltage. In particular, the model is used to define the mechanism of the formation of CU2S on Cu in sulfide-containing brine. The present studies were conducted to provide a scientific basis for estimating the lifetimes of copper canisters in crystalline rock repositories in Sweden for the disposal of high level nuclear waste (HLNW). 

Geringer, J., Taylor, M.L. and Macdonald, D.D. (2012) Predicting the steady state thickness of passive films with the point defect model in fretting corrosion experiments. Proceedings, PRIME 2012 222nd Electrochemical Society Meeting. Pacific Rim Meeting on Electrochemical and Sohd State Science, 7—12 October, Honolulu, Hawaii. 

Other models have also been proposed the point defect model, which is a modified version of the high field model (Lin et al., 1981 Chao et al., 1981) and the place exchange model (Lanyon and Trap-nell, 1955 Sato and Cohen, 1964 Conway and Angerstein-Kozlowska, 1981). 

Passive Nickel and the Point Defect Model The passive state of nickel shows much more complicated behavior than for iron in the same solutions. A minimum of... 

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