the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 10 Jul 2026
The Polymorphism of Snow Crystals: Advances in Understanding Vapor-Phase Ice Growth Dynamics via a Tripartite Coupling Framework
Abstract. Vapor-phase growth of snow crystals generates a striking diversity of morphologies—from simple faceted prisms to complex stellar dendrites, hollow columns, bullet rosettes, and rare trigonal forms—from a single hexagonal ice Ih lattice. This polymorphism emerges from the interplay between temperature-dependent anisotropy in facet-specific attachment kinetics, diffusion-limited vapor transport, and morphological instabilities at the ice–vapor interface. Despite extensive research, no first-principles predictive framework exists; existing models depend on empirical parameterizations of the attachment coefficient α(T) and provide limited insight into how trace atmospheric impurities alter step energetics, quasi-liquid layer (QLL) stability, and growth kinetics. Here we review two decades of advances in laboratory experiments, theory, and simulations since Libbrecht’s 2005 synthesis. We introduce a tripartite coupling framework that unifies the ice crystal, water vapor, and background atmospheric constituents (including impurities). Central to the framework is the Structure-Dependent Attachment Kinetics (SDAK) model, which accounts for habit transitions, edge-sharpening instabilities, trigonal symmetry breaking, and QLL-mediated step dynamics. We discuss extensions to climate microphysics, icephobic surface design, and planetary cryoscience, and identify key remaining challenges and future directions.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-3704', Luis MacDowell, 03 Aug 2026
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RC2: 'Comment on egusphere-2026-3704', Anonymous Referee #2, 18 Aug 2026
Summary and Main Comments:
This article is, ostensibly, a review of ice crystal vapor growth theories and measurements spanning a couple of decades. The authors center the paper on the laboratory and theoretical work of Dr. Kenneth Libbrecht; in some ways, the paper is more of a review paper on Libbrecht’s work. A review paper on the current state of our knowledge of ice crystal vapor growth is needed, especially given that much of that knowledge is spread across several disciplines. I’ve always thought that Libbrecht’s ideas on ice crystal growth need wider exposure especially within the atmospheric community, though many of his papers only reside on a preprint server (arXiv) and have not been thoroughly peer reviewed, so caution is warranted. His ideas are sometimes speculative, but they are creative and may help us move our understanding of ice crystal growth forward. Nevertheless, the current manuscript is both incomplete as a review and is scientifically misleading in some places. The article contains at least one fundamental error, numerous mis-citations of the literature, and it misses important advances in the growth of ice crystals. This manuscript should be rejected. Below I provide comments that back up my recommendation.
Major Comments:
I state above that the manuscript, as a review, is incomplete. While the authors give a brief overview of the influence of attachment kinetics on the growth of isometric (spherical) crystals, much of the latter focus is on Libbrecht’s structure-dependent attachment kinetics (SADK). Libbrecht (2012) points out in one of his arXiv papers that SDAK is speculative and is needed to resolve an apparent paradox. The “critical” supersaturations determined from his experiments were lower for the basal faces of ice at -15C suggesting that columnar crystals should grow instead of the plates that are usually observed at this temperature. The SDAK theory provides a mechanism that not only explains this discrepancy but allows for the sharpening of the rim of a hollowed face. The SDAK theory is interesting, and it may in fact be correct, but it is only one among several theories. For instance, Nelson and Knight’s (1998, J. Atmos. Sci.) measurements produce critical supersaturation values that are exactly the opposite of Libbrecht’s, with prism face values that are the lowest. (It is not presently clear, at least to me, why Libbrecht and Rickerby’s (2015, J. Crystal Growth) and Nelson and Knight’s (1998) critical supersaturation measurements differ from each other. But a review paper should point out these contradictory measurements.) Consequently, the standard theory of faceted growth as applied to hollowed cylinders or hexagonal prisms can explain the growth at -15C. Nelson and Knight (1998) also discuss how growth on the rim of a hollowed face can occur within that theoretical framework. Frank (1982, Contemp. Phys) also discusses how hollowing develops, and how hollowed faces can thin and branches can form. Furthermore, Nelson (2001, Phil. Mag.) goes on to show that column crystals grown near liquid-vapor equilibrium at higher temperatures (near -7C) appear to grow by ledge nucleation near the corners of the basal faces, but by dislocation sources at the centers of the prism faces. The authors discuss the existence of QLLs but never connect it directly to faceted crystals even though there is at least one theory available (Neshyba et al., 2016, J. Geophys. Res.). Since the authors of the present paper spend little time on the theory of faceted growth, readers of this manuscript would miss out on the theories successes (and its failures). The authors also spend little time discussing growth by dislocations, even though this mechanism is probably important for the growth of many atmospheric crystals, especially newly nucleated ones. Indeed, the authors’ historic timeline given in Figure 9 completely ignores some of the major achievements in crystal growth including the development of dislocation theory. I could go on. This article fails as a review, and readers would be much better served reading reviews by Nelson (2001, Phil. Mag.; 2005, Crystal Growth & Design) and Libbrecht (2017, Ann. Rev. Mat. Res.).
There is at least one fundamental error in this manuscript. Under Eq. 13 the authors state that the effects of thermal diffusion/impedance are negligible for typical tropospheric conditions. This conjecture is not true. One wonders if the authors made any attempts to check this assertion by examining papers or textbooks on ice growth in the atmosphere. The 1+g term in their Eq. 13 is indeed small in the upper troposphere, once temperatures fall below about -35C. But the term is very important at higher temperatures. I’ve included a figure (attached as a PDF) showing the influence of that term on the vapor diffusivity (Dv) in the growth rate equations, though I use RT instead of g. As one can see, thermal impedance (RT) becomes important especially when T > -20C. Note that errors between 20 and 40% occur if thermal impedance is ignored for T between -10 and -20C.
Minor Comments:
- Lines 35: You state that modern theories ignore the influence of atmospheric impurities that can shift the habit form, but you don’t provide citations. You also claim that all models use surface models based on empirical fits rather than derivations from interfacial physics. But this is not entirely true. The form of the attachment coefficients is derived from theory, but many of the surface parameters cannot be, or have not been, measured. One consequence is that functions and coefficients are grouped together and determined in aggregate. For instance, Libbrecht’s form for alpha includes an exponential prefactor, A(T), and an scaling supersaturation that is similar to the critical supersaturation in nucleation theory. All the surface physics are now woven into those two coefficients.
- Below line 50: I am not sure what you mean by a “single-factor” or a “phenomenological” treatment. Can you provide examples and citations?
- Figure 1: Is the habit diagram shown at the top of Fig 1 your design? It looks a lot like one from Libbrecht’s papers. If so, you probably need to get permissions from the journal. I noticed this issue in other figures (see below).
- Below Line 70: You gloss over the initial development of facets on a newly formed ice crystal. These don’t just appear and, indeed, the initial formation of facets is a subject that has received comparatively little attention (for images see Takahashi and Mori, 2006, Atmospheric Research; and for theoretical discussions see Nelson and Swanson, 2019, Atmos. Chem. Phys.).
- Below line 85: Pure water does not occur in atmospheric clouds.
- Below line 100: Nucleation does not always lead to hexagonal ice but can be a combination of hexagonal and cubic ice (Shilling et al., 2006, Geophys. Res. Letters).
- Below line 105: Can you provide citations for the facet size limitation before faceting instabilities set in? There are a number of papers on this topic, and it’s never been clear to me that there is a specific size threshold.
- Above line 120: You ignore thermal diffusion in your first sentence because you mistakenly concluded that thermal diffusion can be ignored. And “driven by ice supersaturation”: It’s really the gradient that drives the growth.
- Below line 125: Can you define your crystal radius, R? I don’t think you stated that you assume your crystals are spheres. And by “normal” velocity, I assume you mean dR/dt, or the growth in the direction normal to the crystal surface.
- Below line 135: “impinging molecules must overcome an attachment barrier before incorporating into the lattice.” I presume you mean an attachment energy barrier, and this is a multi-stage process.
- Equation 11: You state that this growth velocity is valid for the mixed kinetic-diffusion regime, but my understanding is that this equation is always valid. One simply couples it with equation 10, and the resulting equation should be valid for the kinetic and diffusion-limited regimes.
- Line 165: Again, thermal diffusion cannot be neglected.
- Section 2.3: It isn’t just the strong temperature dependence of, say, the critical supersaturations that determine the morphology. As I pointed out above, Nelson (2001, Phil. Mag.) showed that the location at which dislocations appear on the surface can influence the development of crystal aspect ratios. In this review, you seem to assume that ledges always form by nucleation and that the source of those ledges is the corners. This is a classic assumption because the supersaturation can maximize at the corners, but it is also possible for the supersaturation to maximize at the facet center (see Plate 2 of Wood et al., 2001, J. Geophys. Res.).
- Your equation 14 is similar, in some ways, to Nelson and Baker’s (1996) Eq. 29: If ledges nucleate near the corners or edges of crystal faces, then theoretically the aspect ratio should asymptotically approach the ratio of the attachment coefficients. This would be a good reference to add here. It is worth keeping in mind that this ratio is only true for crystals that have reached a steady-state aspect ratio under constant environmental conditions. And if the surface location at which the steps form is not the corner/edge, then Eq. 14 and 15 are no longer valid (see Nelson, 2001).
- I’m not sure your Figure 3 is needed.
- In some ways, I like a comprehensive list like Table 1, though mostly for fundamental sources of information. However, some of the sources in your table are misleading (for example Sosso et al’s paper is about ice nucleation in liquids, not ice crystal growth from vapor).
- Above line 220: Can you cite some sources for QLLs? It might be worth pointing out that there is at least one theoretical treatment of faceted growth as mediated by QLLs (Neshyba et al., 2016, J Geophys. Res.).
- Below line 225: You mention the inadequacies of “two-parameter” models and the need for a tripartite treatment. Can you expand on this as I am not sure what you mean. Can you provide citations for two parameter models and explain their limitations. How does your tripartite framework overcome those limitations?
- Below line 230: Bailey and Hallett (2004) is not the best reference to use here. Nelson and Knight (1998) is better.
- Figure 4: You don’t cite this explicitly, but isn’t Fig. 4e from MacDowell’s review paper, Fig. 4d from Llombart’s paper, and Fig. 4f from Hong’s paper? This is not clear in your figure caption. It isn’t clear to me that you can give a blanket statement that “reuse permissions” have been obtained. Your Figure 4a – c also doesn’t seem to match at least some papers on QLL’s and ice surfaces. For example, Nagata et al. (2019, Acc. Chem. Res.) indicate that the first bilayer of ice becomes disordered at temperatures above about -90C whereas the second bilayer becomes disordered once T reaches -16C. QLL films and drops appear only at T > -2C, and the drops appear to be formed kinetically. It’s worth noting that the saturation states needed to from QLL drops are higher than what would be achieved in the atmosphere (see Nagata’s Fig. 4) and so probably never occur on atmospheric ice.
- Eq 16: Do you have a reference for this equation?
- Lines near 265: Where did you come up with the ranges of supersaturations for branching instability? No citation is given. It is also worth mentioning that there is uncertainty whether the Mullins-Sekerka instability is important for dendritic branching. For example, Nelson (2005, pg 1520, below Fig 10) points out that rough surfaces are required for the Mullins-Sekerka instability, so if the dendritic tips are vicinal faces then that mechanism would not apply.
- Line 270: Lamb et al. (2023) discuss the growth of small crystals and focus on attachment kinetic limitations to growth. They do not discuss branching, so this reference is not appropriate.
- Line 270: The sentence beginning “Laboratory observations show…. “ needs a citation.
- Line 280: Sosso discuss ice crystal nucleation in liquid, and so is not an appropriate reference.
- Figure 5: Are the crystal images yours? Or are they from another paper?
- Line 290: “concentration boundary layers”. Here I assume you are referring to the diffusive boundary layer that exists around crystals, which becomes thinner as crystal fall speeds increase. You should clarify this point and provide citations.
- Line 305: Unless I am mistaken, Frank (1982, Contemp. Phys.) also discusses mechanisms that can cause thinning of a hollowed face, and how this can eventually lead to branching.
- Line 325: You discuss the orientation of crystals as they fall, but much has been learned about changes in orientation that could be included.
- Figure 8: These look like Figures from Libbrecht’s papers, and if so, permission to use is probably needed.
- Figure 9: According to this figure, nothing happened in crystal science between Nakaya and Libbrecht? This figure misrepresents the rich history of ice crystal growth research.
- Figure 10: Again, are these crystal images yours or are they from other sources?
- 18: Do you have a citation for this equation?
- Line 635: Why are you citing D’ Acunzi et al (2021) and Gerber et al. (2022)? The first paper is about super liquid repellant coatings, and the second paper is about ice stress. Neither is relevant to the atmosphere.
- Above Line 640: “Current parameterizations rely on simplified habit diagrams and empirical fall speed relationships.” This statement isn’t true and fails the capture the range of ice growth methods used in models. There are a number of methods that predict fall speed relations based on the predicted shapes of growing crystals.
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EC1: 'Comment on egusphere-2026-3704', Daniel Knopf, 24 Aug 2026
Dear Authors,
I agree with the careful assessment of your manuscript by both of the referees. The manuscript does not adhere to the minimum standards of this journal. One cannot readily see how revisions will change this. This review article would need to be completely rewritten to be considered as a new submission. As such, the paper should be withdrawn as both referees recommended a rejection.
Sincerely,
Daniel Knopf
Citation: https://doi.org/10.5194/egusphere-2026-3704-EC1
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In this manuscript the authors attempt to review an important topic in atmospheric physics with no previous documented contributions whatsoever in the field. The effort might have been rewarding for the authors, but they still have a long way of reading and understanding before they are ready to provide a sensible review on atmospheric ice growth. Overall, the result is a disordered and repetitive essay full of vague, misleading, contradictory and unwarranted statements in an elegant wrap that cannot give any useful hint or insight to newcomers in the field. Readers interested in this fascinating topic will do better searching in the original sources, reading a few recent papers on the field or the original 2005 review by Libbrecht cited in this work, the more recent 2017 review in Annu. Rev. Mat. Res. or his book "Snow Crystals" Princeton, 2022 (also available on ArXiv). Alternatively they could also attempt their own review with some dedication and the help of an AI machine. My reccomendation is to reject this manuscript. The quality is sufficient to consider retraction and not have unwary researchers loose their time.
A few comments serve to make the point.
1. The original result of the author's review is a trinity of 'unifying' principles coined under the name of a 'tripartite coupling framework' which unifies two trivial statements, i.e. ice grown from the vapor is about 1. ice and 2. vapor, plus 3. the completely unsupported statement that background atmospheric species are the third crucial element in this 'unified' and 'universal' description. The manuscript is essentially a reiteration of vague or unwarranted statements of this sort, as evident in figure 3, where several of the pictures appear to have little relevance to the statements, but look nice.
2. The manuscript is full of contradictory statements. The introduction states 'no comprehensive first principles framework exists' (line 35), but ends with the statement that the 'tripartite framework ... offers quantiative predictive power ... (including) ... targeted advances in climate modeling" (line 625-630). The manuscript is of this frivolity throughout.
3. The manuscript points to the role of trace gases as a major issue on ice growth, and states this reiteratively without one single source for such claims.
4. On the other hand, several many times citations are misplaced and do not refer to the content:
i. The Barret et al. paper cited in L45 has nothing to do with atomistic simulations. The Hong et al. paper has atomistic simulations as a side issue for the interpretation of AFM expderiments.
ii. Llombart et al cited in L100 is not an adequate source of evidence for hexagonal crystals.
iii. Kling et al and Sibley et al cited in L235 do not address lowering of kinetic barriers at all.
iv. None of the references cited in L240 adress kinetic roughening, far less at -40 C.
v. Sosso et al cited in L280 do not address QLL dynamics at branch tips at all.
5. Aside a lack of any depth, there are several misleading statements: a) the suggestion that the nucleation of ice is mostly due to homogeneous deposition growth from the vapor, which is actually the less expected of all mechanisms (this is visible right away in figure 2, together with the suggestion that an ice embryo at relevant atmospheric temperatures to the Nakaya diagram could be a boat hexagon). b) the statement that growth in the atmosphere is driven by surface free energy minimization (line 100), in complete contradiction with Librecht 's main claim. c) the statement that attachment coefficients serve for the design of icephobic materials (L650), which are all about ice-substrate interactions , d) trace gas can poison the ice surface and prevent growth.e) the statement (L690) that Machine Learned potentials have provided insight into step propagation, which has never been dealt with in the few published ML potential simulations of QLL to date (of course, no citation to this claim)....
Credit attribution is not appropriate. All of the figures on ice crystals are from Kent Libbrecht but credit missing mostly everywhere in the captions.