Classical nucleation theory diffusion

The classic nucleation theory is an excellent qualitative foundation for the understanding of nucleation. It is not, however, appropriate to treat small clusters as bulk materials and to ignore the sometimes significant and diffuse interface region. This was pointed out some years ago by Cahn and Hilliard [16] and is reflected in their model for interfacial tension (see Section III-2B). 

Classical nucleation theory (CNT) shows that I is a product of the probability of diffusion and that of formation of a critical nucleus [1,4],... 

The onset of crystallization in the Vitreloy 1 alloy shown in Fig. 1.3 was based on the detection of a crystalline volume fraction of 10 " by conventional methods. There are two different curves for the onset of crystallization in Fig. 1.3. For the solid curve, which was obtained using classical nucleation theory, the effective diffusion constant was taken to be proportional to the reciprocal of the viscosity, which was considered to be of the VFT form. The dotted curve was obtained by the use of an Arrhenius form of expression for the effective diffusion constant, which was found to fit better at lower temperatures near Tg (Masuhr et al. 1999). This discussion shows that for understanding the kinetics of crystallization in multi-component alloys considerable improvisation becomes necessary. 

Figure 3. Experimental test of classical nucleation theory. Comparison between experimental (exp) and theoretical (th) nucleation rates for droplet condensation from n-nonane vapor. Subscript int denotes the integrated theoretical nucleation rate along the height of the thermal diffusion cloud chamber [37].

In a comprehensive treatment of the nucleation and crystallization of glass, it would be necessary to consider spinodal separation into two phases by a diffusion-like process without initial formation of surfaces, and consequently without the surface energy barrier encountered in classical nucleation theory ( ). However, this type of nucleation does not appear to enter into the crystallization we have observed in the Ba0-B203-Al203 system. 

The nucleation theory just described is referred to as classical nucleation theory. It relies on the capillarity approximation, in which crystallites of microscopic size are treated as if they are macroscopic, and in which the kinetics is described as the stepwise attachment of single molecules across the crystal-melt interface. In fact, this approximation may not be valid under realistic conditions. A small crystallite may not achieve bulk properties at its center, and its interface may be so strongly curved that the planar value Ysl no longer applies. The interface may be diffuse rather than sharp, so that the description of the kinetics as resulting from addition of solid particles one after another may not be valid instead, a collective fluctuation may result in the simultaneous incorporation of a larger number of molecules in a loosely structured crystallite. 

This is a much more flexible theory than classical nucleation theory (which allows the free energy to depend only on a single quantity, the crystallite radius). Here, the interface can be diffuse instead of sharp, the crystallite can be strongly curved, and the order parameters can attain values different from the bulk at the center. At the same time, the nonclassical density functional approach goes over smoothly to the classical theory for large enough crystallites. Comparisons of the predictions of the classical... 

For most CVD reactions, the supersaturation is so high that calculated values of r are of atomic dimensions. For such reactions, the classical theory is not appropriate, and detailed atomic treatments must be considered. Because of the interesting fundamental questions underlying nucleation and the important applications of thin films, interest in modeling adsorption, surface diffusion, and nucleation has been considerable. These efforts are described in several, well-documented reviews (61, 75-78). 

The above trends are illustrated in Figure 3. It shows thermal diffusion cloud chamber measurements of homogeneous nucleation rates for n-nonane. The fact that isotherms aie parallel to the 45" line along which experiments and classical theory agree demonstrates that the dependence on supersaturation is correctly captured by the theory. The fact that the isotherms don t collapse on this line demonstrates that the theory underpredicts nucleation rates at low temperatures, and overpredicts them at high temperatures. 

A numerical analysis of Equation 5.4 with Ag(c) in the form of Equation 5.6 was done for different widths of the diffusion zone down to fC = 0.5 nm. We considered two different cases (i) without shape optimization (nucleus being a cube 21 X 21 X 21 and the shape parameter is fixed attp = ) and (ii) with shape optimization, that is, for each nucleus volume tp is determined to minimize AG. The calculated results demonstrate, in both cases, that even at a very narrow interdiffusion width the nucleation barrier is rather low, about 20kBT (Figure 5.4), and the critical size 21 o. of the nucleus amounts to 0.45 nm, which nearly coincides with the value of classical theory and implies practically immediate nucleation. 

In order to monitor the mechanical properties in relation to the microstructure, the knowledge of the precipitation state at the end of a thermo-mechanical treatment is of prime importance. In this purpose, Arcelor develops models that allow for the prediction of the influence of the process parameters on the state of precipitation. The model Multipreci, developed at IRSID is one of them. It (Hedicts the precipitation kinetics of mono- and di-atomic particles in ferrite and austenite as a function of the time-temperature history. It is based on the classical theories for diffusive phase transformation and treats simultaneously the nucleation, growth and ripening phenomena. The state of precipitation that is predicted includes the particle size distribution, their number and volume fraction. From these values, the effect of the precipitates on the mechanical properties can be calculated. 

Based on the classical theory on foam nucleation, macroscopic properties such as solubility S, diffusivity D, gas concentration C, surface tension o, temperature and degree of supersaturation are the parameters controlling the nucleation rate J. There are several equation proposed for J, but they can be summarized in the following general expression. ... 

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