Two-step crystal nucleation kinetics: Solution of the master equation

The master equation of two-step nucleation is numerically solved with parameters characterizing isothermal homogeneous nucleation of ice crystals and water droplets in steam below the water freezing temperature. The time dependences of the crystal and droplet size distributions and nucleation rates as well as of the nucleation rate of ice crystals in the droplets are determined at different constant supersaturations. The stationary values of these nucleation rates are obtained in a wide supersaturation range. It is found that when the crystallization of the water droplets is slower than their growth, the stationary rate of two-step crystal nucleation is orders of magnitude lower than the corresponding stationary rate of one-step crystal nucleation. As to the stationary nucleation rate of the water droplets, it is practically unaffected by the ice crystals nucleating and growing in both the droplets and the steam. Finally, the delay times of the different nucleation processes are determined and it is found that at both low and high supersaturations the delay time of the two-step crystal nucleation is negative because of the presence of high initial peak in the nonstationary rate of this process. The results obtained provide unprecedented mechanistic insight into the kinetics of two-step crystal nucleation.