Nucleation density functional approach

Up to this point, all the analysis of nucleation presented has been based on the classical nucleation theory of Section II. To go beyond it, and indeed to begin to test its validity, a new approach is needed. In this section we summarize the results of a recent density functional approach to nucleation, and show that the results differ in quantitative detail but not in overall character from the predictions of the classical theory. 

Gunton subsequently extended Lunger sfield theory of nucleation to the liquid-solid transition, within the context of the density functional approach of Harrowell and Oxtoby. In this way they were able to calculate the preexponential factor in the nucleation and obtained results in agreement with qualitative expectations from the classical theory. 

Ten Wolde and Frenkel [171] have made the very interesting observation that this hidden transition can nonetheless profoundly influence the crystallization behavior of the system. Fluids that are in the vicinity of this submerged critical point display substantial density fluctuations, just as they do when near a usual critical point. The crystallization mechanism in these instances proceeds by a route in which the fluid fluctuates to a solid-like density before arranging itself into a crystal form. This is in contrast to a mechanism in which the crystal first nucleates into a very small crystal, which then grows as it encounters additional fluid molecules. This understanding can contribute to the difficult art of crystallizing proteins. In fact, successful crystallizations have been known to be associated with a fluid opalescence that previously was not considered to be in any way related to the same effect seen in critical fluids. Density-functional approaches have since been applied and found to support the ten Wolde-Frenkel hypothesis [172]. 

Kusaka, I., Wang, Z.-G., and Seinfeld, J. H. (1995a) Ion-induced nucleation A density functional approach, J. Chem. Phys. 102, 913-924. 

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... 

In classical nucleation theory the density of a cluster is assumed to be equal to the bulk liquid density, and the free energy of the surface (surface tension) is taken as that of a bulk liquid with a flat interface. A cluster is then completely defined by its number of molecules or its radius. In general, there is no reason why the density at the center of a cluster should equal the bulk liquid density, nor is there any reason that the density profile should match that at a planar interface (Figure 10.8). The density p r) should not be constrained other than to require that it approach the bulk vapor density at large distances from the cluster. Whereas the condition for the critical nucleus in classical theory is that the derivative of the free energy with respect to radius r be equal to zero, when p is allowed to vary with position r, the free energy is now a function of that density. The free energy has a minimum at the uniform vapor density and a second, lower minimum at the uniform liquid density. Between these two minima lies a saddle point at which the functional derivative of the free... 

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