Modeling of Precipitation Kinetics

In Equation (4.18) the reaction of interstitially dissolved metal species has been described using the size-distribution function f(r, t) of precipitates, which has been introduced into

image146 image147 Подпись: (4.20)

gettering simulations at different levels of accuracy. In order to fully grasp precipitation processes, one needs to describe particle nucleation, growth and coarsening, the latter of which is known as Ostwald ripening. A frequently used approximation (see, e. g., [73, 139]) is to describe f (r, t) as a monodispersive distribution, i. e. a fixed density Np of precipitates that yields


Подпись:d^p R ^M

dt = p Np

where Vp is the time-dependent volume of the precipitate and ^M the volume of metal atoms in the precipitating phase. This approximation ignores precipitate nucleation, which is phenomenologically introduced by the precipitate density but fails to describe the incubation period associated with the formation of supercritical nuclei (for an instruc­tive description, see, e. g., [140]). Precipitate dissolution during gettering is qualitatively described but Ostwald-ripening effects are ignored. The latter will slow down dissolution kinetics so that the monodispersive approximation underestimates the time necessary to dissolve precipitates during gettering. Please note, that the use of Ham’s theory [141] in its simplest form further simplifies the description to the case of a fixed precipitate radius that is valid for later stages of precipitate growth in still supersaturated solutions.

Подпись: df d t Подпись: dn |[g {nA) Подпись: d(n,t)]f image153 Подпись: (4.22)

An improved description is achieved by using the Fokker-Planck equation [142, 143] to describe the time dependence of the size-distribution function. Using the notation in [144] one gets

where f = f (n, t) is the size-distribution function using the number n of metal atoms in a precipitate as a variable, whereas g (n, t) and d (n, t) denote the growth and disso­lution rate, respectively. It should be noted that in spatially inhomogeneous systems-as is the case for gettering processes-all quantities entering into Equation (4.22) are posi­tion dependent. According to [145] this approach is valid for n > 10, while for smaller precipitates a rate equation approach is more appropriate. Senkader et al. [144] use a combination of both methods to describe oxygen precipitation. For larger precipitate sizes the Fokker-Planck equations are a valid approach to describe Ostwald-ripening phenomena during gettering annealing. A similar model has been implemented in order to simulate gettering and precipitation of iron in silicon [146, 147].