Transient nucleation induction time from the birth-death equations

For the set of finite-difference equations of Becker-Döring an exact formula for the induction time, which is expressed in terms of rapidly convergent sums, is presented. The form of the result is particularly amenable for analytical study, and the latter is carried out to obtain approximations of the exact expression in a rigorous manner and to assess its sensitivity to the choice of the nucleation model. The induction time, find, is found to be governed by two main nucleation parameters, Φ_*/kT, the normalized barrier height, and g_*, the number of molecules in the critical cluster. The ratio of these two parameters provides an assessment of the importance of discreteness effects. We study the exact expression in both the continuous (g_* → ∞) and the asymptotic (Φ_*/KT → ∞) limits. Asymptotic results for find are compared with those previously reported from simulation studies as well as with find obtained numerically from the exact expression in the present study. Also, the accuracy of the Zeldovich equation, which is produced in the continuous limit, is discussed for several nucleation models.