Nucleation is of great concern in many cases—for example, the production of artificial rainfall and the synthesis of advanced amorphous alloys. Although exact solutions have been well known to both homogeneous nucleation and heterogeneous nucleation occurring on a large flat container wall, yet in more general situations the actual nucleation takes place around finite-sized heterogeneous particles. The understanding of nucleation in such situations requires a more extended model which considers the size effect of nucleating agents. Partially motivated by our research on bulk metallic glasses, we construct such a geometric model. Also we derive an exact solution to the model and discuss briefly its physical implications. A previously presumed relation between the critical energy barrier $\left(E_c\right)$ and the volumetric Gibbs free energy of the critical nucleus $\left(G_c\right)$—i.e., $E_c=\frac{1}{2}{G}_c$—is found to be not true for general cases, although it is correct for the limiting cases.

• Received 25 March 2005

DOI:https://doi.org/10.1103/PhysRevB.72.052101