We here review mostly experimental and some computational work devoted to nucleation in amorphous ices. In fact, there are only a handful of studies in which nucleation and growth in amorphous ices are investigated as two separate processes. In most studies, crystallization temperatures Tx or crystallization rates RJG are accessed for the combined process. Our Review deals with different amorphous ices, namely, vapor-deposited amorphous solid water (ASW) encountered in many astrophysical environments; hyperquenched glassy water (HGW) produced from μm-droplets of liquid water; and low density amorphous (LDA), high density amorphous (HDA), and very high density amorphous (VHDA) ices produced via pressure-induced amorphization of ice I or from high-pressure polymorphs. We cover the pressure range of up to about 6GPa and the temperature range of up to 270K, where only the presence of salts allows for the observation of amorphous ices at such high temperatures. In the case of ASW, its microporosity and very high internal surface to volume ratio are the key factors determining its crystallization kinetics. For HGW, the role of interfaces between individual glassy droplets is crucial but mostly neglected in nucleation or crystallization studies. In the case of LDA, HDA, and VHDA, parallel crystallization kinetics to different ice phases is observed, where the fraction of crystallized ices is controlled by the heating rate. A key aspect here is that in different experiments, amorphous ices of different "purities" are obtained, where "purity" here means the "absence of crystalline nuclei." For this reason, "preseeded amorphous ice" and "nuclei-free amorphous ice" should be distinguished carefully, which has not been done properly in most studies. This makes a direct comparison of results obtained in different laboratories very hard, and even results obtained in the same laboratory are affected by very small changes in the preparation protocol. In terms of mechanism, the results are consistent with amorphous ices turning into an ultraviscous, deeply supercooled liquid prior to nucleation. However, especially in preseeded amorphous ices, crystallization from the preexisting nuclei takes place simultaneously. To separate the time scales of crystallization from the time scale of structure relaxation cleanly, the goal needs to be to produce amorphous ices free from crystalline ice nuclei. Such ices have only been produced in very few studies.
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