The Nakaya snow crystal morphology diagram, showing different types of snow crystals that grow in air at atmospheric pressure, as a function of temperature and water vapour supersaturation relative to ice. The water saturation line gives the supersaturation of supercooled water, as might be found within a dense cloud. Note that the morphology switches from plates ( T ≈ − 2 C) to columns ( T ≈ − 5 C) to plates ( T ≈ − 15 C) to predominantly columns ( T < − 30 C) as temperature is decreased. Temperature mainly determines whether snow crystals will grow into plates or columns, while higher supersaturations generally produce more complex structures. 

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... We describe a comprehensive model for the formation and morphological development of atmospheric ice crystals growing from water vapor, also known as snow crystals. Our model derives in part from empirical measurements of the intrinsic ice growth rates as a function of temperature and supersaturation, along with additional observations and analyses of diffusion-driven growth instabilities. We find that temperature-dependent conformational changes associated with surface melting strongly affect layer nucleation dynamics, which in turn determines many snow-crystal characteristics. A key feature in our model is the substantial role played by structure-dependent attachment kinetics, producing a growth instability that is largely responsible for the formation of thin plates and hollow columnar forms. Putting these elements together, we are able to explain the overall growth behavior of atmospheric ice crystals over a broad range of conditions. Although our model is complex and still incomplete, we believe it provides a useful framework for directing further investigations into the physics underlying snow crystal growth. Additional targeted experimental investigations should better characterize the model, or suggest modifications, and we plan to pursue these investigations in future publications in this series. Our model also suggests new avenues for the continued exploration of ice surface structure and dynamics using molecular dynamics simulations. Laboratory observations of snow crystals dating back to the 1930s have revealed a complex dependence of growth morphologies on temperature and supersaturation [1, 2]. Under common atmospheric conditions, for example, ice crystals typically grow into thin plate-like forms near -2 C, slender columns and needles near -5 C, thin-walled hollow columns near -7 C, very thin plates again near -15 C, and columns again below -30 C. In addition, morphological complexity generally increases with increasing supersaturation at all temperatures. These observations are often summarized in the well-known Nakaya morphology diagram, and one example is shown in Figure 1 [1]. Extensions to lower temperatures, as well as more detailed morphological studies, can be found in the literature [3, 4, 5]. Although the overall features and morphological transitions seen in the Nakaya diagram have been well established empirically, a basic physical explanation of why snow crystals exhibit this growth behavior has been surprisingly elusive. In particular, the fact that snow crystals alternate between plate-like and columnar forms as a function of temperature has been an outstanding problem for nearly 75 years. The purpose of this paper is to present a new model for ice growth from ...

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... key role in the case of solidification from the liquid phase, while for solidification from gaseous precursors the anisotropy in the surface attachment kinetics is typically the more dominant factor [1, 9]. Numerical models of diffusion-limited growth were developed about the same time as solvability theory, including front-tracking and phase-field techniques [10, 11]. These methods have enjoyed considerable success in reproducing dendritic structures arising during solidification from the melt, such as in metallurgical systems or ice growth from liquid water. These systems work well compu- tationally in part because the material anisotropies are typically quite small, with surface energy differences of perhaps a few percent between faceted and non-faceted surfaces. For such systems the growth structures are generally smooth and free of sharp corners or edges, and numerical models tend to be robust and stable. In growth from the vapor phase, anisotropies in the surface attachment kinetics are often very large, resulting in dendritic structures that are not smooth with continuous derivatives, but are instead strongly faceted with sharp edges. In this case numerical instabilities can be problematic, so considerably more care is required to produce stable growth models [12]. As a result, the usual numerical techniques used for studying diffusion-limited growth have not yet been able to produce satisfactory snow crystal structures from reasonable physical inputs. In 2008-9, Gravner and Griffeath developed cellular automata (CA) techniques that avoided the numerical instabilities that affected other methods [13, 14]. These CA models have generated full three-dimensional dendritic structures that reproduced many characteristics of natural snow crystals, including growth forms that are both branched and faceted, with sharp edges. Cellular automata models incorporating more physically derived rules have since been demonstrated [15], and with suitable inputs it now appears possible, at least in principle, to realize a numerical model that can accurately reproduce snow crystal growth rates and morphologies at all temperatures and supersaturations. Analysis of diffusion-limited growth using these theoretical and computational tools has yielded numerous insights into the dynamics of structure formation. For example, solvability theory nicely explains why the tip velocity of a growing dendritic structure depends linearly on supersaturation for solidification from vapor, while a quadratic dependence on undercooling is typical for growth from the liquid phase [1, 9]. In addition, scaling relationships in diffusion-limited growth models provide an explanation for the increase in structural complexity that accompanies decreasing vapor diffusion rates [15, 16]. For snow crystal growth, these theoretical considerations generally explain why morphological complexity increases with supersaturation, crystal size, and background gas pressure. Thus the observed variation along the supersaturation axis on the morphology diagram in Figure 1 is fairly well understood, at least at a qualitative level. Producing accurate model crystals using sensible input physics over a range of conditions has not yet been accomplished, and some unusual dendritic snow crystal structures may be quite challenging to reproduce [17]. Nevertheless, diffusion-limited growth – the underlying physical mechanism responsible for the formation of much snow crystal structure – is reasonably well understood, and satisfactory computational algorithms describing this process are available. The ability to make high-fidelity numerical models of dendritic growth is only the first step, however, since models require inputs. For the snow crystal case, one of the most important physical inputs comes from the many-body interactions that determine how water molecules are incorporated into the crystalline lattice, referred to as the surface attachment kinetics. For a rough surface, this in- corporation is essentially instantaneous for all molecules that strike the surface. But the ...