Random distribution active site model for ice nucleation in water droplets
Modeling of ice nucleation in water droplets in clouds is essential for predicting cloud lifetime and climate change. We propose a stochastic model describing both nucleation time distributions under isothermal conditions and nucleation temperature distributions under cooling conditions. The model assumes the presence of active sites of single activity inducing heterogeneous nucleation. It is termed random distribution active site model. The active sites are assumed to spread randomly over water droplets according to the Poisson distribution. Monte Carlo simulations are performed to generate both the nucleation time distributions and the nucleation temperature distributions. Analytical solutions are also obtained for the two limiting cases i and ii. Case i assumes uniform distribution of active sites and case ii assumes no presence of active sites (homogeneous nucleation only). These two analytical solutions are confirmed to be in good agreement with the Monte Carlo simulations. By fitting the analytical solutions to literature data on immersed mode heterogeneous nucleation in water droplets (B. J. Murray, et al., Atmos. Chem. Phys., 2011, 11, 4191), nucleation kinetic parameters are determined, and then heterogeneous nucleation rates are calculated. The model for the uniform distribution active sites case is shown to have the same mathematical structure as the single component stochastic model (B. J. Murray, et al., Chem. Soc. Rev., 2012, 41, 6519). The relation of the proposed model to the singular model (G. Vail, J. Atmos. Sci., 1971, 28, 402) is also discussed; the number of active sites per unit surface of immersed particles ns in the singular model is shown to be equal to the heterogeneous nucleation rate per unit surface area of immersed particles Jhet divided by the decreasing rate of a dimensionless temperature in the proposed model. The parameter ns is not the number of the active sites per unit surface of immersed particles as claimed. The effects of cooling rate, immersed particle concentration and droplet volume on nucleation temperatures are also discussed theoretically.