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

摘要

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=r~(max)(t) and approaches a universal shape exp[ζ-e~ζ], with ζ~∝ 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 zther applications might be anticipated due to universality of the nucleation pulse technique.