Transient nucleation driven by solvent evaporation

Abstract

We theoretically investigate homogeneous crystal nucleation in a solution containing a solute and a volatile solvent. The solvent evaporates from the solution, thereby continuously increasing the concentration of the solute. We view it as an idealized model for the far-out-of-equilibrium conditions present during the liquid-state manufacturing of organic electronic devices. Our model is based on classical nucleation theory, taking the solvent to be a source of the transient conditions in which the solute drops out of solution. Other than that, the solvent is not directly involved in the nucleation process itself. We approximately solve the kinetic master equations using a combination of Laplace transforms and singular perturbation theory, providing an analytical expression for the nucleation flux, predicting that (i) the nucleation flux lags slightly behind a commonly used quasi-steady-state approximation, an effect that is governed by two counteracting effects originating from the solvent evaporation: while a faster evaporation rate results in an increasingly larger influence of the lag time on the nucleation flux, this lag time itself we find to decrease with increasing evaporation rate, (ii) the nucleation flux and the quasi-steady-state nucleation flux are never identical, except trivially in the stationary limit and (iii) the initial induction period of the nucleation flux, which we characterize with a generalized induction time, decreases weakly with the evaporation rate. This indicates that the relevant time scale for nucleation also decreases with increasing evaporation rate. Our analytical theory compares favorably with results from numerical evaluation of the governing kinetic equations.