Growth law parabolic

It should be emphasized that n.. and JPS, and therefore c and T, refer to the condition at the pore tip. The dissolution valence and the temperature can be assumed to be independent of pore depth. This is not the case for the HF concentration c. Because convection is negligible in macropores, the mass transport in the pore occurs only by diffusion. A linear decrease in HF concentration with depth and a parabolic growth law for the pores according to Pick s first law is therefore expected, as shown in Fig. 9.18 a. The concentration at the pore tip can be calculated from the concentration in the bulk of the electrolyte c, the pore length l, the diffusion coefficient DHf (Section 1.4) and the flow of HF molecules FHf. which is proportional to the current density at the pore tip ... 

If the rate is controlled by diffusive mass transfer (Figure 1-1 lb) and if other conditions are kept constant, then (i) the growth (or dissolution) distance is proportional to the square root of time, referred to as the parabolic growth law (an application of the famous square root law for diffusion), (ii) the concentration in the melt is not uniform, (iii) the concentration profile propagates into the melt according to square root of time, and (iv) the interface concentration is near saturation. For the rate to be controlled by diffusion in the fluid, it cannot be stirred. 

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. 

Equation (6.41) states that the parabolic growth law also applies to the total thickness, and the relative thickness A[Pg.154]

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. 

Integration of this growth-rate expression then yields the parabolic growth law... 

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. 

Depending on the relative rates of diffusion of these species, the reaction can take place at either the AO/AB2O4 or the B2O3/AB2O4 interface. When diffusion is slower than the rate of reaction, the thickness of the product layer follows a parabolic growth law like that observed in Figure 5.20 for NLA1204. 

In this case, the most likely mechanism is the counterdiffusion of cations, i.e., the mechanism of Fig. 3.3c, where the electroneutrality is maintained by the coupling of flux of the cations. When the formation rate of the product is controlled by diffusion through the layer of the product, the thickness of the product layer will follow a parabolic growth law, which is given by ... 

This is known as lander equation, which has two oversimplifications that limit its applicability and the capability to predict the rates of most chemical reactions. Firstly, the parabolic growth law for the thickness of the product layer is only valid for one-dimensional reaction across a planar boundary, but not suflhcient for the reactions involving spherical particles. In other words, this assumption is only valid for the initial stage of the reaction when y r. Second, the changes in molar volumes of the reactants and the products are not taken into account. To address this problem, a more comprehensive equation should be used, which is [51, 52] ... 

For 2D grain growth, kinetic models generally predict a parabolic growth law of the form [13]... 

In the ternary system, the values of all thermodynamic variables will be fixed on all phase boundaries if local equilibrium is maintained and if the partial pressure of oxygen, as well as P and r, is fixed. Therefore, the defect gradient, and thus the particle flux, in every phase of the reaction product will be inversely proportional to the corresponding thickness of the phase. This, then, results in a parabolic growth law for each phase of the reaction product, as follows from the discussion of section 6.2.1. Let (p) be an index which denotes the phase, and let Ic be the practical rate constant. Then ... 

Here is the parabolic oxidation constant (kg m s ). Integration of (9.17) with the condition ffjoxd = 0 at f = 0, yields parabolic growth law ... 

According to the parabolic growth law (9.17), the rate of oxidation is inversely proportional to the mass of the oxide present, moxd- If the density of the oxide Poxd is independent of the layer thickness, we can set ... 

This equation has the same mathematical form as the parabolic growth law (9.18). The oxidation constant is equal to ... 

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