Asymptotic and NumericalStudies of the Becker - Doring Model for Transient Homogeneous Nucleation
2006, v.12, Issue 2, 341-365
Transient homogeneous nucleation is studied in the limit of large critical sizes. Starting from pure monomers, three eras of transient nucleation are characterized in the classic Becker - Doring kinetic equations with the Turnbull - Fisher discrete diffusivity. After an initial stage in which the number of monomers decreases, many clusters of small size are produced and a continuous size distribution is created. During the second era, nucleii are increasing steadily in size in such a way that their distribution appears as a wave front advancing towards the critical size for steady nucleation. The nucleation rate at critical size is negligible during this era. After the wave front reaches critical size, it ignites the creation of supercritical clusters at a rate that increases monotonically until its steady value is reached. Analytical formulas for the transient nucleation rate and the time lag are obtained that improve classical ones and compare very well with direct numerical solutions. In addition, we propose and solve numerically a modified Becker - Doring model having a discrete diffusivity proportional to the area of a spherical cluster with $k$ monomers for small $k$ (as in the Turnbull - Fisher case) and to the cluster radius for large $k$ (as in the case of diffusive growth of clusters).
Keywords: kinetics of first order phase transitions,homogeneous nucleation,Becker - Doring equations,singular perturbation,asymptotic theory