Curvature and Nucleation

The bottom-up formation of small (nano) particles begins with nucleation, clustering together of atoms or molecules. Because of the strong curvature of the nuclei, the Kelvin (or Ostwald) effect plays an important role. The Kelvin effect implies that the ratio c R)/c (R = oo) or p R)/p (R = =°) increases rapidly as R decreases to very small values, as can be seen in Table 6.1. In view of this, phenomena such as supersaturation, supercooling, and superheating are involved in the onset of phase separation, that is, the formation of nuclei of the uew phase. 

The Gibbs energy of forming a liquid droplet of radius a is 

When p(R = a) p(R = the first term on the right-hand side of Equation 6.25 makes AG 0 for sufficiently large values of a if a is smaller, the second term dominates over the first one, resulting in AG 0. 

FIGURE 6.9 Gibbs energy of formation of a droplet as a function of the droplet radius. 

Evaporation and crystallization of a liquid as well as precipitation from a solution can be treated in the same way as described for vapor condensation. For instance, for the precipitation process this results in 

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