Growth laws

In most practical cases (and at moderate voltages) the high-field growth law can control film growth, say up to only a maximum of 10 nm, as at this thickness the field strength effects become even less important than film growth due to diffusion of vacancies or ions. 

The above rate law has been observed for many metals and alloys either anodically oxidized or exposed to oxidizing atmospheres at low to moderate temperatures—see e.g. [60]. It should be noted that a variety of different mechanisms of growth have been proposed (see e.g. [61, 62]) but they have in common that they result in either the inverse logaritlnnic or the direct logarithmic growth law. For many systems, the experimental data obtained up to now fit both growth laws equally well, and, hence, it is difficult to distinguish between them. 

If a compact film growing at a parabolic rate breaks down in some way, which results in a non-protective oxide layer, then the rate of reaction dramatically increases to one which is linear. This combination of parabolic and linear oxidation can be tenned paralinear oxidation. If a non-protective, e.g. porous oxide, is fonned from the start of oxidation, then the rate of oxidation will again be linear, as rapid transport of oxygen tlirough the porous oxide layer to the metal surface occurs. Figure C2.8.7 shows the various growth laws. Parabolic behaviour is desirable whereas linear or breakaway oxidation is often catastrophic for high-temperature materials. 

This is the Wilson-Frenkel rate. With that rate an individual kink moves along a step by adsorbing more atoms from the vapour phase than desorbing. The growth rate of the step is then simply obtained as a multiple of Zd vF and the kink density. For small A/i the exponential function can be hnearized so that the step on a crystal surface follows a linear growth law... 

For a small driving force this growth law is indeed slower than the Wilson-Frenkel law (33) with Fwf but incomparably much larger than that of the nucleation process on faceted surfaces, (24), with V exp(—c/A/.i), where c is a positive constant. Therefore, the... 

The scaling arguments given here for two-dimensional growth patterns can be extended formally in a straightforward fashion to three dimensions. For dendritic structures this seems to be perfectly permissible since the basic growth laws are rather similar in two and three dimensions [117,118] ... 

All reactions in which charge is transported through a film of reaction product on the metal surface —the film may or may not be rate determining (e.g. parabolic, logarithmic, asymptotic, etc. or linear growth laws, respectively). 

If K = 1 K, a = 0.25 nm, and z = 3, X = 30nm at 300 K, so that for a film 1 nm thick, the field increases the rate of growth by a factor of about 10 The term in the growth law due to the field, namely exp (K/X), is large only when X is small. Because of this a thin oxide film can form even at low temperatures where the ordinary rate of entry of ions into the oxide, is negligible. As the film thickens, the factor exp /X) decreases rapidly, and the rate of growth soon falls to such a low value that, for practical purposes, oxidation has ended. 

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) . 

Evans has shown how the effect of internal stresses in growing films may have various effects that can lead to any one of the first three growth laws referred to above. 

Conway BE, Barnett B, Angerstein-Kozlowska H, Tilak BV. 1990. A surface-electrochemical basis for the direct logarithmic growth law for initial stages of extension of anodic oxide films formed at noble metals. J Chem Phys 93 8361-8373. 

Validation of Experimental Growth Laws Through Field Data Interpretation... 

McMurray, P. H. and J. C. Wilson, Growth Laws for the Formation of Secondary Ambient Aerosols Implications for Chemical Conversion Mechanisms, J. Geophy. Res.. 88(C9) 5101-5108 (1983). 

The kinetics of the nonconserved order parameter is determined by local curvature of the phase interface. Lifshitz [137] and Allen and Cahn [138] showed that in the late kinetics, when the order parameter saturates inside the domains, the coarsening is driven by local displacements of the domain walls, which move with the velocity v proportional to the local mean curvature H of the interface. According to the Lifshitz-Cahn-Allen (LCA) theory, typical time t needed to close the domain of size L(t) is t L(t)/v = L(t)/H(t), where H(t) is the characteristic curvature of the system. Thus, under the assumption that H(t) 1 /L(t), the LCA theory predicts the growth law L(t) r1 /2. The late scaling with the growth exponent n = 0.5 has been confirmed for the nonconserved systems in many 2D simulations [139-141]. 

In a real system there will be several clusters growing simultaneously. At first the clusters are separated, but as they grow, they meet and begin to coalesce (see Fig. 10.5), which complicates the growth law. For the case of circular growth considered here, the Avrami theorem [4]... 

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