Parabolic growth kinetics

High-temperature oxidations forming fairly thick ( > 1000 A) product layers exhibit parabolic growth kinetics... 

Consider the results obtained by E.M. Tanguep Njiokep et al. with Al-Mg couples as an example of diffusional parabolic growth kinetics. Consideration will mainly be restricted to a temperature of 400°C at which two intermetallic compounds, Al3Mg2 (also designated as Al8Mg5) and Al12Mg17, are known to exist and which is well below both eutectic temperatures.142 193207... 

Fig. 16. Concentration profile and growth kinetics for the case of the diffusion of uncharged defects, (a) Linear concentration profile (b) parabolic growth kinetics.

G. Borchardt and G. Strehl. On Deviations from Parabolic Growth Kinetics in High Temperature Oxidation. In H. Bode (ed.). Materials Aspects in Automotive Catalytic Converters, pp. 106-116, Weinheim, 2002. DGM - Deutsche Gesellschaft ftir Materialkunde e.V, Wiley-VCH. International Conference Materials Aspects in Automotive Catalytic Converters , 3 October 2001, Munich, Germany. 

By use of the proper experimental conditions and Ltting the four models described above, it may be possible to arrive at a reasonable mechanistic interpretation of the experimental data. As an example, the crystal growth kinetics of theophylline monohydrate was studied by Rodriguez-Hornedo and Wu (1991). Their conclusion was that the crystal growth of theophylline monohydrate is controlled by a surface reaction mechanism rather than by solute diffusion in the bulk. Further, they found that the data was described by the screw-dislocation model and by the parabolic law, and they concluded that a defect-mediated growth mechanism occurred rather than a surface nucleation mechanism. 

Growth kinetics of two chemical compound layers in a binary heterogeneous system have been theoretically treated, from a diffusional viewpoint, by V.I. Arkharov,1 46 K.P. Gurov el al.,22 B. Schroder and V. Leute,52 A.T. Fromhold and N. Sato,53 G.-X. Li and G.W. Powell,55 and other investigators. Diffusional considerations predict that (z) both layers must occur simultaneously and (ii) the thickness of each of them as well as their total thickness should increase parabolically with passing time. 

A system of differential equations of this type appears to have been first proposed by J. Loriers in 1949 (see Ref. 13) to describe paralinear growth kinetics of two oxide layers. The term paralinear growth, being a combination of the words parabolic and linear, means that some initial portion of the time dependence of the total thickness or mass of two compound layers is almost parabolic and then there is a gradual transition to linear kinetics. 

It should be emphasised that in general initial portions of the x-t and y - t dependences are not parabolic. Nonetheless, in the case under consideration some initial portion of the time dependence of the total thickness of both layers is close to a parabola. This portion (from 0 to about 400 s) can be described with fairly good accuracy by the parabolic equation x = 2k t, where k = (8 2)xl0 m s. From a formal viewpoint, it can therefore be concluded (especially if only the dependence of the total thickness or mass of both compound layers upon time is analysed) that the parabolic growth law gradually transforms into the linear growth law, whereas the layer-growth kinetics are in fact somewhat more complicated. 

Both trajectories are seen in Fig. 2.14 to asymptotically tend with increasing time to a straight line corresponding to a constant ratio of the layer thicknesses. Whenever these are sufficiently close to this line, the parabolic growth law becomes a good approximation for both layers and therefore can be employed to treat the experimental kinetic data. 

The layer-growth kinetics were found to be parabolic for both compounds (Fig. 2.18), indicative of diffusion control. This is an expectable result since the layer thickness varied from about 10 pm to 300 pm for the Al12Mg17 intermetallic compound and from about 80 pm to more than 900 pm for the Al3Mg2 intermetallic compound. Diffusional constants were calculated using parabolic equations of the type x2 = 2k t. The temperature dependence of the diffusional constants was found to obey the Arrhenius relation ... 

This binary system is worth further investigation, especially in the region of non-parabolic layer-growth kinetics. Marker experiments are also desirable, with inert markers embedded in both intermetallic layers. 

Note that even in those cases where multiple compound layers were present at the A-B interface, two layers were dominating. For example, G. Hillmann and W. Hofmann and O. Taguchi et al. observed the formation of all six intermetallics shown on the equilibrium phase diagram in the reaction zone between zirconium and copper, with two Cu-rich compounds occupying more than 90 % of the total layer thickness and layer-growth kinetics deviating from a parabolic law. When investigating... 

If the growth regimes of all the layers are reaction controlled (in the theoretical definition given in Chapter 1) at least with regard to one component, then they can in principle grow simultaneously whatever their number. Note that in this case the layer-growth kinetics can hardly be expected to obey a parabolic law. This is characteristic of very thin compound layers, at most a few hundreds of nanometres thick, if not less. 

As the thickness of the ApBq layer increases with passing time, its growth rate must gradually decrease. If kom + k0A2 b, kom hm/x, k()A2 k A2/x and (kW + k t2)/x b, then the growth kinetics of the ApBq layer will be almost parabolic. The abundance of necessary conditions to be satisfied indicates that this is a rather rare case. Strictly speaking, the initial portion of the layer thickness-time dependence can in general hardly be expected to be either linear or parabolic. 

In contrast to the Fe2Al5 layer (see Fig. 1.2), the M0AI4 layer is seen to have relatively even interfaces with both initial phases. In the case of the Mo-saturated aluminium melt, its growth kinetics follows the parabolic law jc2 = 2k t (Fig. 5.14). In the 750-850°C range the temperature dependence of the growth-rate constant, ku is described by the equation ... 

Fig. 5.19. Sulphidation kinetics of molybdenum from the gaseous phase at 750°C and a sulphur vapour pressure of 4 Pa (0.03 mm Hg). Growth kinetics of the M0S2 layer are initially linear and then parabolic. According to the experimental data by B.S. Lee and R.A. Rapp.364...

It seems relevant to remind once again that in the case of formation of a single-phase compound layer, the reverse (parabolic-to-linear) transition is impossible. From a physicochemical viewpoint, it is only possible during the simultaneous occurrence of two or more compound layers, as is indeed observed experimentally. Parabolic-to-linear growth kinetics are thus indicative of the formation of multiple layers of oxides, nitrides, sulphides, etc., even though some of them may be unindentifiable due to their extremely small thickness. 

Nevertheless, in this book the number of the theoretically substantiated kinetic equations, for the experimentalist to use in practice, appears to exceed that resulting from purely diffusional considerations. Whether the experimentalist will be pleased with such an abundance of equations is a wholly different question. Still, for many researchers in the field it is so tempting to employ the only parabolic relation and then to discuss in detail the reasons for (unavoidable and predictable) deviations from its course. Note that unlike diffusional considerations where each interface is assumed to move according to the square root of the time, in the framework of the physicochemical approach the layer-growth kinetics are not predetermined by any additional assumptions, except basic ones, but immediately follow in a natural way from the proposed mechanism of the reaction-diffusion process. 

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. 

Making the additional approximation that Rg Rg. = 2.5 Rg, the grain growth kinetics can be shown to be parabolic, like that of Burke and Turnbull. 

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