  In this work a phenomenological stochastic differential equation is proposed
to model the time evolution of the radius of a pre-critical molecular cluster
during nucleation (the classical order parameter). Such a stochastic
differential equation constitutes the basis for the calculation of the
(nucleation) induction time under Kramers' theory of thermally activated escape
processes. Considering the nucleation stage as a Poisson rare-event, analytical
expressions for the induction time statistics are deduced for both steady and
unsteady conditions, the latter assuming the semiadiabatic limit. These
expressions can be used to identify the underlying mechanism of molecular
cluster formation (distinguishing between homogeneous or heterogeneous
nucleation from the nucleation statistics is possible) as well as to predict
induction times and induction time distributions. The predictions of this model
are in good agreement with experimentally measured induction times at constant
temperature, unlike the values obtained from the classical equation, but
agreement is not so good for induction time statistics. Stochastic simulations
truncated to the maximum waiting time of the experiments confirm that this fact
is due to the time constraints imposed by experiments. Correcting for this
effect, the experimental and predicted curves fit remarkably well. Thus, the
proposed model seems to be a versatile tool to predict cluster size
distributions, nucleation rates, (nucleation) induction time and induction time
statistics for a wide range of conditions (e.g. time-dependent temperature,
supersaturation, pH, etc.) where classical nucleation theory is of limited
applicability.
