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Homogeneous nucleation of ice in supercooled water droplets is a stochastic process. In its classical description, the growth of the ice phase requires the emergence of a critical embryo from random fluctuations of water molecules between the water bulk and ice-like clusters, which is associated with overcoming an energy barrier. For heterogeneous ice nucleation on ice-nucleating surfaces both stochastic and deterministic descriptions are in use. Deterministic (singular) descriptions are often favored because the temperature dependence of ice nucleation on a substrate usually dominates the stochastic time dependence, and the ease of representation facilitates the incorporation in climate models. Conversely, classical nucleation theory (CNT) describes heterogeneous ice nucleation as a stochastic process with a reduced energy barrier for the formation of a critical embryo in the presence of an ice-nucleating surface. The energy reduction is conveniently parameterized in terms of a contact angle alpha between the ice phase immersed in liquid water and the heterogeneous surface. This study investigates various ice-nucleating agents in immersion mode by subjecting them to repeated freezing cycles to elucidate and discriminate the time and temperature dependences of heterogeneous ice nucleation. Freezing rates determined from such refreeze experiments are presented for Hoggar Mountain dust, birch pollen washing water, Arizona test dust (ATD), and also nonadecanol coatings. For the analysis of the experimental data with CNT, we assumed the same active site to be always responsible for freezing. Three different CNT-based parameterizations were used to describe rate coefficients for heterogeneous ice nucleation as a function of temperature, all leading to very similar results: for Hoggar Mountain dust, ATD, and larger nonadecanol-coated water droplets, the experimentally determined increase in freezing rate with decreasing temperature is too shallow to be described properly by CNT using the contact angle alpha as the only fit parameter. Conversely, birch pollen washing water and small nonadecanol-coated water droplets show temperature dependencies of freezing rates steeper than predicted by all three CNT parameterizations. Good agreement of observations and calculations can be obtained when a pre-factor beta is introduced to the rate coefficient as a second fit parameter. Thus, the following microphysical picture emerges: heterogeneous freezing occurs at ice-nucleating sites that need a minimum (critical) surface area to host embryos of critical size to grow into a crystal. Fits based on CNT suggest that the critical active site area is in the range of 10-50 nm(2), with the exact value depending on sample, temperature, and CNT-based parameterization. Two fitting parameters are needed to characterize individual active sites. The contact angle alpha lowers the energy barrier that has to be overcome to form the critical embryo at the site compared to the homogeneous case where the critical embryo develops in the volume of water. The pre-factor beta is needed to adjust the calculated slope of freezing rate increase with temperature decrease. When this slope is steep, this can be interpreted as a high frequency of nucleation attempts, so that nucleation occurs immediately when the temperature is low enough for the active site to accommodate a critical embryo. This is the case for active sites of birch pollen washing water and for small droplets coated with nonadecanol.

If the pre-factor is low, the frequency of nucleation attempts is low and the increase in freezing rate with decreasing temperature is shallow. This is the case for Hoggar Mountain dust, the large droplets coated with nonadecanol, and ATD. Various hypotheses why the value of the pre-factor depends on the nature of the active sites are discussed.