Transformations of the distribution of nuclei formed in a nucleation pulse: Interface-limited growth

A typical nucleation-growth process is considered: a system is quenched into a supersaturated state with a small critical radius r^- _* and is allowed to nucleate during a finite time interval t_n, after which the supersaturation is abruptly reduced to a fixed value with a larger critical radius r⁺_*. The size-distribution of nucleated particles f (r,t) further evolves due to their deterministic growth and decay for r larger or smaller than r+, respectively. A general analytic expressions for f (r,t) is obtained, and it is shown that after a large growth time t this distribution approaches an asymptotic shape determined by two dimensionless parameters, λ related to t_n, and = r⁺_*/r^-_*. This shape is strongly asymmetric with an exponential and double-exponential cutoffs at small and large sizes, respectively, and with a broad near-flat top in case of a long pulse. Conversely, for a short pulse the distribution acquires a distinct maximum at r= rmax (t) and approaches a universal shape exp [ξ- e^ξ], with ξ α r r^max, independent of the pulse duration. General asymptotic predictions are examined in terms of Zeldovich-Frenkel nucleation model where the entire transient behavior can be described in terms of the Lambert W function. Modifications for the Turnbull-Fisher model are also considered, and analytics is compared with exact numerics. Results are expected to have direct implementations in analysis of two-step annealing crystallization experiments, although other applications might be anticipated due to universality of the nucleation pulse technique.