Parabolic law of growth

This is the parabolic law of growth which evidently applies to the high-temperature oxidation of rinc. 

The integrated form is x = Kt -i- C, or x = D Vt, where D is the Diffusion Coefficient. Eiquation 4.5.3. is called the Parabolic Law of Diffusion. If the growth of a phase can be fitted to this equation, then it is likely that the primary reaction mechanism involves simple diffusion. 

Solution of the differential equation obtained by combining Eqs (5.8.20) and (5.8.21) yields the Wagner parabolic law of film growth,... 

The boundary kinetics (stages 2-4) may result in changing the constant of solid-state reaction rate and deviation from the parabolic law of phase growth controlled by stages 1 and 5 [8]. 

Further development of Velikanov s concept of the PFC proved its usefulness not only for electrolysis of chalcogenide melts. For example, representation of the oxide film on a metal surface as PFS easily explains the Tamman s parabolic law for growth of such film in the air corrosirMi process of metals. Formation of a film of intermediate compounds on the surface of electrode explains many regularities of long-term electrolysis of the melts cmitaining compounds of polyvalent metals (see [7] for more details). 

If Q is the volume of the oxide per metal atom, the rate of growth, dA /d/, is equal to / Q. Thus from equation 1.179 we derive the parabolic 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). 

The volume ratio (see Section 1.9) for cuprous oxide on copper is 1 7, so that an initially protective film is to be expected. Such a film must grow by a diffusion process and should obey a parabolic law. This has been found to apply for copper in many conditions, but other relationships have been noted. Thus in the very early stages of oxidation a linear growth law has been observed (e.g. at 1 000°C) . 

The possible employment of beryllium in nuclear engineering and in the aircraft industry has encouraged considerable investigation into its oxidation characteristics. In particular, behaviour in carbon dioxide up to temperatures of 1 000°C has been extensively studied and it has been shown that up to a temperature of 600°C the formation of beryllium oxide follows a parabolic law but with continued exposure break-away oxidation occurs in a similar fashion to that described for zirconium. The presence of moisture in the carbon dioxide enhances the break-away reaction . It has been suggested that film growth proceeds by cation diffusion and that oxidation takes place at the oxide/air interface. ... 

Kinetic expressions for appropriate models of nucleation and diffusion-controlled growth processes can be developed by the methods described in Sect. 3.1, with the necessary modification that, here, interface advance obeys the parabolic law [i.e. is proportional to (Dt),/2]. (This contrasts with the linear rate of interface advance characteristic of decomposition reactions.) Such an analysis has been provided by Hulbert [77], who considers the possibilities that nucleation is (i) instantaneous (0 = 0), (ii) constant (0 = 1) and (iii) deceleratory (0 < 0 < 1), for nuclei which grow in one, two or three dimensions (X = 1, 2 or 3, respectively). All expressions found are of the general form... 

At high temperature and a higher partial pressure of oxygen (1 < P(02) < 20 torr), the rate of growth of the FeO layer follows the parabolic rate law. The rate of formation of FeO is determined by the rate of diffusion of Fe2+, but the rate of diffusion of O2- determines the rate at which the thickness of Fe203 increases. 

If the transport process is rate-determining, the rate is controlled by the diffusion coefficient of the migrating species. There are several models that describe diffusion-controlled processes. A useful model has been proposed for a reaction occurring at the interface between two solid phases A and B [290]. This model can work for both solids and compressed liquids because it doesn t take into account the crystalline environment but only the diffusion coefficient. This model was initially developed for planar interface reactions, and then it was applied by lander [291] to powdered compacts. The starting point is the so-called parabolic law, describing the bulk-diffusion-controlled growth of a product layer in a unidirectional process, occurring on a planar interface where the reaction surface remains constant ... 

Fig. 18.3 Plots of the growth laws of oxidation a) parabolic, b) rectilinear, c) quasi-rectilinear, d) logarithmic (West, 1980, with permission).

The parabolic law means that the growth (or dissolution or oxidation) distance is proportional to the square root of time,... 

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. 

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. 

It should be noted that in many works the degree of saturation of liquid B with the dissolving component A is not taken into account, and even in cases where a pure solvent is used, the experimental data are treated using a linear, a parabolic, a logarithmic or some other dependence in order to establish the so-called growth law of a chemical compound layer. It is clear, however, that both the academic and practical value of such laws is... 

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 ... 

It should be noted that at this stage the rate of increase of the total mass of the layers would be less than that which could be expected on the basis of a parabolic law. Therefore, if the kinetic data are formally treated using a power law, its exponent must in all likelihood be greater than 2. The cubic law may presumably hold in this region of growth of compound layers, though experimental values of the exponent are rarely observed to be constant and equal to 3. Most frequently, its value varies in the range 2.0-3.5.7... 

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