Parabolic kinetic behavior

The Wagner theory [10] for a coupled diffusion and migration (also known as ambipolar diffusion) mass transfer during oxidation of metals and alloys is briefly described. This theory treats the parabolic kinetic behavior of high temperature oxides given by eq. (10.19) with n = 1/2. 

Parabolic kinetics" refers to the observation that the concentration of a species released to an aqueous solution by alteration of a primary mineral plots as a linear function of the square root of time. Figure 1 (data from 5) illustrates this. Numerous workers (e.g., 5-11) have observed this behavior. 

At least four different explanations have been proposed to account for parabolic kinetics. The oldest and best established is the "protective-surface-layer" hypothesis. Correns and von Englehardt (6) proposed that diffusion of dissolved products through a surface layer which thickens with time explains the observed parabolic behavior. Garrels ( 12, 1 3) proposed that this protective surface consists of hydrogen feldspar, feldspar in which hydrogen had replaced alkali and alkaline earth cations. Wollast (j>) suggested that it consists of a secondary aluminous or alumino-silicate precipitate. In either case, a protective surface layer explains parabolic kinetics as follows If the concentration of any dissolved product at the boundary between the fresh feldspar... 

Figure 1 Example of parabolic kinetics showing linear behavior of silica concentration vs. the square root of time, from Ref. 5.

The third proposed explanation for parabolic kinetics is that dissolved products may be released from the mineral surface linearly, but that non-linear precipitation of secondary minerals from solution accounts for the non-linear concentration vs. time behavior (16). 

Studies on the oxidation behavior of dense SiCNO ceramic materials revealed the formation of a dense and continuous oxide layer with a sharp oxide/ceramic interface and parabolic kinetics from 800 °C to 1400 °C (Chollon, 2000). The parabolic constants and the activation energies were found to be similar with the values obtained for SiC and SijN. These results may be explained if oxidation of SiCNO PDCs is attributed to an oxidation mechanism involving the formation of an intermediate silicon oxynitride layer, which is also found for silicon nitride ceramics (Chollon, 2010). 

For the same type of catalyst we have observed in a recirculation laboratory reactor multiplicity, periodic and chaotic behavior. Unfortunately, so far we are not able to suggest such a reaction rate expression which would be capable of predicting all three regimes [8]. However, there is a number of complex kinetic expressions which can describe periodic activity. One can expect that such kinetic expressions combined with heat and mass balances of a tubular nonadiabatic reactor may give rise to oscillatory behavior. Detailed calculations of oscillatory behavior of singularly perturbed parabolic systems describing heat and mass transfer and exothermic reaction are apparently beyond, the capability of both standard current computers and mathematical software. 

A is the sum of the solvent and intramolecular reorganization energies, and AG = F(A 0 - A 0J) is the standard electrochemical Gibbs energy of the electron transfer from x = a to x = b. Parabolic dependence of AG on AG was demonstrated [viii]. Electrochemical behavior including the kinetic analysis of various ET systems was reviewed [ix]. A special type of the ET reaction is the deposition of a metal at ITIES, e.g., the deposition of Au particles by the interfacial reaction between AUCI4 in 1,2-dichloroethane and Fe(CN)6 in water [x]. 

Slow Growth Regime (a < 5). The surface is atomically smooth and growth is controlled by the movement of steps emerging from dislocations. Slow growth kinetics and parabolic dependence on supersaturation results, and growth behavior is as predicted by the Burton-Cabrera-Frank (Burton, W.K., et al. 1951) model... 

Figure 9. Schematic of the behavior of the parabolic rate constant as a function of oxygen pressure for the oxidation of copper. The rate increases /I/n = I/S, see Equation 20] until the formation of CuO occurs. At that point the chemical potentials of oxygen are fixed both at the Cu-Cu O and at the CugO—CuO interface, and the kinetics become independent of external oxygen pressure (13).

These data have to be treated with some caution, because many results are obtained from treatment in vacuum without a protective scale present. The kinetics of the reaction have been described as parabolic for a number of cases. However, they depend also on physical conditions like the wetting behavior of the Uquid metal or a formed melt [64],... 

The Reynolds number in microreaction systems usually ranges from 0.2 to 10. In contrast to the turbulent flow patterns that occur on the macroscale, viscous effects govern the behavior of fluids on the microscale and the flow is always laminar, resulting in a parabolic flow profile. In microfluidic reaction systems, where the characteristic length is usually greater than 10 pm, a continuum description can be used to predict the flow characteristics. This allows commercially written Navier-Stokes solvers such as FEMLAB and FLUENT to model liquid flows in microreaction channels. However, modeling gas flows may require one to take account of boundary sUp conditions (if 10 < Kn < 10 , where Kn is the Knudsen number) and compressibility (if the Mach number Ma is greater than 0.3). Microfluidic reaction systems can be modeled on the basis of the Navier-Stokes equation, in conjunction with convection-diffusion equations for heat and mass transfer, and reaction-kinetic equations. 

Hence, the oxide thickness becomes x = f(T). It is known that the oxide kinetics of many metals and alloys show parabolic behavior. However, other type of behavior is possible and eq. (10.12) is generalized and modeled as an empirical relationship given by... 

The kinetics of oxidation of tantalum in pure oxygen have been studied at temperatures up to 1400°C and at pressures ranging from less than 1 to over 40 atm (0.10-4.05 MPa). The reaction is initially parabolic, with a transformation to linear rate after a period of time. Increasing the temperature not only increases the rate of oxidation, but also decreases the time before the reaction changes from parabolic to linear behavior. Above about 500°C and pressures from 10 mm Hg to 600 psi (1333 Pa to 4.13 MPa), the transition occurs almost immediately. From 600 to 800°C, the oxidation shows a pronounced increase in rate with pressure above 0.5 atm (0.05 MPa). At 1300°C and 1 atm oxygen pressure, tantalum oxidizes rapidly and catastrophically, but at 1250°C, the metal oxidizes linearly for a short time, then catastrophically. Unlike tantalum-oxygen reactions, however, tantalum-air reactions do not exhibit catastrophic oxidation at temperatures as high as 1400°C. 

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