Parabolic oxidation law

Figure 1-13 Comparison of (a) parabolic oxidation law for fayalite oxidation at 1043 K in air and (b) linear oxidation law for fayalite oxidation at 1303 Kina mixture of CO and CO2 (Mackwell, 1992). The vertical axis x is the thickness of the oxidized layer.

This time dependence of is known as parabolic oxidation law. 

A parabolic oxidation law is assumed so that simple relations can be derived. Jost (28) as well as Valensi (29) and Wagner (quoted in 23) have discussed more general assumptions. These relations, which have been derived elsewhere (1) in somewhat modified and generalized form, are based on the similarity of the defect and the distributions in each individual partial surface layer with variable value, and on the assumption that at the boundary of two partial layers a value of (jl and jug, respecitvely, prevails which is independent of the overall value, as, for example, in oxides near the equilib -... 

B/A is called the linear rate constant. Once a thick oxide forms, the oxidation relationship reduces to a parabolic oxidation law,... 

Solving for t yields the well-known parabolic oxidation law for passive oxidation processes ... 

FIGURE 5.17 The parabolic oxidation law plotted in terms of (a) thickness as the dependent axis and (b) time as the dependent axis. A crossover from a reaction-limited linear growth rate to a diffusion-hmited square-root growth rate occurs at a critical oxide thickness when the two limiting rate processes are equal. 

Problem 5.8. A silicon wafer is being oxidized in an atmosphere of pure oxygen at 1200 K. The thickness of the oxide, x (in pm), as a function of time t (in seconds) is given by the parabolic oxidation law ... 

In certain circumstances even the parabolic rate law may be observed under conditions in which the oxide is porous and permeated by the oxidising environment". In these cases it has been shown that it is diffusion of one or other of the reactants through the fluid phase which is rate controlling. More usually however the porous oxide is thought to grow on the surface of a lower oxide which is itself growing at a parabolic rate. The overall rate of growth is then said to be paralinear - and may be described by the sum of linear and parabolic relationships (see equations 1.197 and 1.198). 

If the PBR is less than unity, the oxide will be non-protective and oxidation will follow a linear rate law, governed by surface reaction kinetics. However, if the PBR is greater than unity, then a protective oxide scale may form and oxidation will follow a reaction rate law governed by the speed of transport of metal or environmental species through the scale. Then the degree of conversion of metal to oxide will be dependent upon the time for which the reaction is allowed to proceed. For a diffusion-controlled process, integration of Pick s First Law of Diffusion with respect to time yields the classic Tammann relationship commonly referred to as the Parabolic Rate Law ... 

Sometimes crystal growth, dissolution, or oxidation is said to follow a linear growth law or a parabolic growth law. The linear law means that the thickness of the crystal depends linearly on time. 

Figure 5.6 Rate laws for formation of oxide films, (a) Parabolic rate law. (b) Effect of film cracking (successive parabolic segments), (c) Limiting case of (6). (d) Logarithmic rate law.

These assumptions, however, oversimplify the problem. The parent (A,B)0 phase between the surface and the reaction front coexists with the precipitated (A, B)304 particles. These particles are thus located within the oxygen potential gradient. They vary in composition as a function of ( ) since they coexist with (A,B)0 (AT0<1 see Fig. 9-3). In the Af region, the point defect thermodynamics therefore become very complex [F. Schneider, H. Schmalzried (1990)]. Furthermore, Dv is not constant since it is the chemical diffusion coefficient and as such it contains the thermodynamic factor /v = (0/iV/01ncv). In most cases, one cannot quantify these considerations because the point defect thermodynamics are not available. A parabolic rate law for the internal oxidation processes of oxide solid solutions is expected, however, if the boundary conditions at the surface (reaction front ( F) become time-independent. This expectation is often verified by experimental observations [K. Ostyn, et al. (1984) H. Schmalzried, M. Backhaus-Ricoult (1993)]. 

The oxidation of iron at temperatures between 500 and 700 K was also studied by means of backscattered conversion electron spectroscopy by Simmons et al. (235). An example of these typical upside-down backscatter spectra is shown in Fig. 36. The authors found that the kinetics for oxide formation followed a parabolic rate law and that the resulting oxide formed at these low temperatures was nonstoichiometric Fe304. 

This completes our development of the thick-film parabolic growth law. This particular theory has been presented in some detail because it is an extremely important domain of metal oxidation. In addition, it provides an excellent example of the way the coupled-currents approach [10,11] can be used to obtain oxide growth kinetics and built-in voltages in thermal oxidation. 

Table VIII lists the same quantities for the parabolic rate law for columbium, tantalum, titanium, zirconium, and iron. The K values show that the resistance to oxidation increases in the series columbium, tantalum, zirconium, titanium, iron. This is borne out by a comparison of the AF values for these oxidations. Although the oxidation of tantalum needs the highest energy of activation E, it is actually not as resistant to oxidation as, for example, iron, since the entropy difference for the tantalum oxidation is not sufficiently negative to make the overall potential barrier as high as that existing for the oxidation of iron. These...

FOLLOWING A SHORT introduction dealing with the relationship between diffusion process and field transport phenomena in tarnishing layers on metals and alloys, the mechanism of oxidation of iron is discussed. Epitaxy plays an important role on the gradient of the concentration of lattice defects and, therefore, on the validity of the parabolic rate law. Classical examples of metal oxidation with a parabolic rate law are presented and the various reasons for the deviation observed are elucidated on the systems Iron in CO/CO2 and CU2O in <>2. In addition, the oxidation of alloys with interrupted oxide-metal interfaces is treated. Finally, attention is focussed on the difficulties in explaining the low temperature-oxidation mechanism. 

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