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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. 

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...

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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 ...

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... Snow crystal morphology diagram showing types of snow crystals that grow at different temperatures and humidity levels. Picture taken fromLibbrecht, 2012. ...
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... Skeletal biomineral morphologies will self-assemble in such a setting due to Berg effect, where supersaturation over a flat face is low at the center and high near the edges, causing first "hopper like" then true single-crystalline ordered dendrites, disordered polycrystalline side branches, and finally disordered polycrystalline dendrites and dense branching morphologies as the kinetic forcing becomes more extreme. Sunagawa (1999) shows that it is a general rule that growth rate anisotropy will determine the forms of polyhedral crystal, as seen in the development of snow crystals (Libbrecht, 2012), where solid hexagonal plates crystallize at low supersaturation, whereas increasing the supersaturation leads to dendritic growth and flower-like morphologies with six petals. Beck and Andreassen (2012) report similar kinetic control on vaterite formation, where low levels of supersaturation lead to the growth of hexagonal, plate-like crystals while increasing kinetic forcing force causes a shift of the particle growth mechanisms toward dendritic growth. ...
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... Simulated snowflakes show excellent agreement with experimental observations. [16][17][18][19]29 Their growth satisfies the microscopic solvability theory, [30][31][32][33] consistently with experiments. 34 MODEL Snowflakes growth in supersaturated water vapour was simulated using a phase field approach in three dimensions, based on a methodology developed in. ...
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... Often the focus is on the c axis orientation between crystallites, so the a axis orientation has been less relevant. As focus shifts to fundamental issues such as face-specific ice growth kinetics (55,(62)(63)(64)(65)(66)(67) and its dependence on molecular-level structure, a axis orientation becomes more critical. ...
... Small perturbations tip the balance of which is the most stable, and hence largest, face (26,27). This delicate balance is thought to explain the dendritic shape of snowflakes (55,(62)(63)(64)(65)84). Although a comprehensive model is still lacking, it is likely that snowflake growth (growth at the solid-vapor interface) is kinetically controlled. ...
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... Equation 4 has been modified to include radiative effects in the context of cirrus modeling (Sölch & Kärcher 2010), although they have not been reported to be significant for contrails. The Nakaya crystal morphology diagram depicts the empirically observed shape (which Equation 4 cannot predict) that ice crystals assume with a given homogeneous temperature and supersaturation (Libbrecht 2005(Libbrecht , 2012. Much has been learned in the past few years about how one could go about predicting the shape, and for this we refer the reader to the given references. ...
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... Our understanding of these data is quite crude, in part because our overall understanding of the many subtleties of crystal growth dynamics is somewhat rudimentary, especially in a material like ice that exhibits substantial surface premelting. In [9] we attempted to construct a comprehensive physical picture of ice growth from water vapor, connecting the growth measurements shown in Figure 1 with related morphological observations. Since our present purpose is to examine the correspondence between growth from water vapor and from liquid water, we will focus on the growth from vapor at T = −2 C, which is the highest temperature for which we have data in Figure 1. ...
... Both the basal and prism data suggest a smooth transition from solid/liquid growth to solid/vapor growth near the melting point. At lower temperatures the QLL becomes thinner and eventually disappears entirely, resulting in the complex growth behavior described in more detail in [9]. ...
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We examine ice crystal growth from water vapor at temperatures near the melting point, when surface premelting creates a quasiliquid layer at the solid/vapor interface. Recent ice growth measurements as a function of vapor supersaturation have demonstrated a substantial nucleation barrier on the basal surface at these temperatures, from which a molecular step energy can be extracted using classical nucleation theory. Additional ice growth measurements from liquid water as a function of supercooling exhibit a similar nucleation barrier on the basal surface, yielding about the same molecular step energy. These data suggest that ice growth from water vapor and from liquid water are both well described by essentially the same underlying nucleation phenomenon over a substantial temperature range. A physical picture is emerging in which molecular step energies at the solid/liquid, solid/quasiliquid, and solid/vapor interfaces create nucleation barriers that dominate the growth behavior of ice over a broad range of conditions. Since the step energy is an equilibrium quantity, just as surface melting is an equilibrium phenomenon, there exists a considerable opportunity to use many-body simulations of the ice surface structure and energetics at equilibrium to better understand many dynamical aspects of ice crystal growth.
... One reason for this state of affairs is that the surface attachment kinetics depend in detail on the molecular structure of the ice surface, including surface melting, which is itself not well understood. In addition, we have recently found that the attachment kinetics depend on the mesoscale surface structure, further complicating our picture of ice crystal growth [4]. ...
... This method affords many advantages, including: 1) small crystals can be observed, which is important for minimizing diffusion effects and measuring surface attachment kinetics; 2) interferometric techniques can be used for making precise measurements of growth rates; 3) aligning a crystal facet to the substrate gives a well-defined growth morphology relative to the substrate, and 4) the supersaturation is set by the temperature difference between the growing crystal and a nearby water vapor reservoir, which can yield exceptionally good determinations of this important quantity. The substrate method was used to make the most accurate measurements to date of the attachment coefficient for ice growth over a range of conditions [6], as well as exploring aspects of structure-dependent attachment kinetics [4]. Overall this method is perhaps the most useful for making precise measurements of ice growth rates in well-defined conditions. ...
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... In [4] we further presented a new and comprehensive physical model that begins to explain the overall structure of the morphology diagram, in particular the observed changes in growth morphology as a function of temperature. The SDAK instability plays a central role in this model, in that it connects the intrinsic growth rates of faceted surfaces to the observed morphological changes with temperature. ...
... We find clear evidence for SDAK effects on the basal facets, suggesting that the SDAK instability is largely responsible for the formation of thin-walled hollow columnar crystals near this temperature. These results support the model in [4], and strongly support the hypothesis that SDAK effects play an important role in determining the growth morphologies of atmospheric ice crystals. ...
... 2 Intrinsic Growth Rates at -5.15 C Following [4], we define the intrinsic growth rates of the basal and prism surfaces as the growth rates of infinite, clean, dislocation-free, faceted ice surfaces in near equilibrium with pure water vapor at a fixed temperature. We parameterize the surface growth velocities using v = α surf v kin σ surf , where v is the perpendicular growth velocity, v kin (T ) is a temperature-dependent "kinetic" velocity derived from statistical mechanics, and σ surf is the water vapor supersaturation relative to ice at the growing surface. ...
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We present experimental data demonstrating the presence of structure-dependent attachment kinetics (SDAK) in ice crystal growth from water vapor near -5 C. Specifically, we find that the nucleation barrier on the basal edge of a thin-walled hollow columnar crystal is approximately ten times smaller than the corresponding nucleation barrier on a large basal facet. These observations support the hypothesis that SDAK effects play an important role in determining the growth morphologies of atmospheric ice crystals as a function of temperature.
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