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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), by Cao Zexian (Institute of Physics, Chinese Academy of Sciences)

Snow is one of the small-scale solid forms of water. Snowflakes are generally hexagonal, with D6 symmetry, but each snowflake is unique. How can snowflakes have different shapes while maintaining D6 symmetry has always been a puzzling question. The first person to study the formation mechanism of snowflakes was Kepler, who gave the three laws of planetary motion, and brought the ball-dense packing model of solids.

A remarkable feature that distinguishes the earth from other planets is that 70% of its surface is covered with water. Assuming that the earth's surface is evenly covered by water, the average water depth can reach 2700 meters. The physical condition of the earth's surface is precisely that the solid-liquid-gas phase of water can coexist in a corner (that is, the temperature-pressure condition of the earth's surface is just near the three-phase point of water! This feature is of second to none in understanding the origin of life on earth, as well as in understanding the physics created by human beings (figure 1). Keep in mind that all the properties of water are abnormal and cannot be judged by common sense. The solid phase of water, as far as bulk ice is concerned, is known to have 16 crystalline phases, three of which are lighter than liquid water. Thankfully, ice, which is common in nature, Ih phase, is lighter than water-if not, you won't have a chance to skate on the river unless the river is completely frozen from the bottom to the top. By the way, the homogeneous nucleation temperature of water is about 232K (minus 41 ℃). In other words, the freezing of water at 0 ℃ is because the water is dirty, it is in contact with other substances, or because it is turbulent. The solid phase of water is also small and not so well crystallized, including frost,rime, snow, rime, hail, soft hail (graupel), sleet (sleet) and so on. Among them, snow is the most beautiful, and there is a saying of snowflakes in Chinese. The swirling snowflakes have aroused the reverie of many people.

Figure 1. The common scene on the surface: the solid-liquid-gas three-phase coexistence of water in a small range. Snowflakes are generally flaky, millimeter-sized and visible to the naked eye. One of the characteristics of snowflakes is that different snowflakes are generally hexagonal. Han Ying said in the Foreign Legend of Korean Poems in the Western Han Dynasty that "there are five more plants and six snowflakes". The first half of this sentence is wrong, but the second half is correct. In the later literature, the theory of snowflake six can be found everywhere, but it is actually very vague. Liu-out, what kind of six-out method is it? Further, we can ask, why?

The earliest documented study of the shape and formation mechanism of snowflakes is the German scientist Kepler (Johannes Kepler, 1571-1630), who left us three laws of planetary motion to explore the mysteries of God because of the claim that "I am better than you humans". As early as 1611, Kepler published a 24-page pamphlet, hexagonal Snow (de nive sexangula) (figure 2), which tried to explain the hexagonal shape of snowflakes with the accumulation model of small balls. The accumulation of small balls is certainly not enough to explain the hexagonal morphology of snowflakes. However, Kepler's book is the first to use the ball accumulation model to understand the atomic structure of matter, especially crystals. It can be said that Kepler's research sowed the seeds of crystallography-the original geometry of crystals can be explained by the accumulation of small balls. In addition, Kepler's research raises an important mathematical problem, which is today known as the Kepler conjecture, that is, hexagonal dense packing is the densest packing for identical spheres. Kepler had such a great influence on crystallography that in 1981 someone wrote a classic paper, "Pentagon Snow" (Alan L. Mackay, De Nive Quinquangula, Krystallografiya, Vol. 26,910,919 (1981)). In 1984, quasicrystals (quasicrystal) with fifth symmetries were discovered.

Fig. 2 Kepler's hexagonal Snow and its Ball accumulation Model A prerequisite for understanding the shape and formation mechanism of snowflakes is to know what snowflakes look like. However, even in the cold north of China, it is difficult to observe and record the shape of snowflakes. Snowflakes are very small (millimeter size) and melt quickly. Therefore, although there is a saying of "six snowflakes" in the literature of our ancestors, it is very difficult to communicate with others what snowflakes are like. If you want to talk about snowflakes, you must first paint and take pictures of snowflakes. The first snowflake photo is believed to have been taken by a German Johann Heinrich Ludwig Fl ö gel (1834-1918) in 1879 (figure 3).

Figure 3. Frog's snowflake photos taken in 1879 take snowflake photography seriously as a career. American Wilson Alwyn Bentley (1865-1931) (figure 4). Bentley was born in 1865 in Jericho, Vermont, USA, which is a famous snow belt with an annual snowfall of up to 300cm. At the age of 15, Bentley received a birthday present from his mother-a small microscope, an ordinary act of family warmth that led to a major event in the history of science. Bentley likes photography, and the heavy snow in his hometown has aroused his curiosity. At some point, he had a fervent desire to take a picture of the snowflake. In 1885, 19-year-old Bentley added a microscope to his camera and obtained his first picture of snowflakes on January 15 (figure 5). Obviously, Bentley's snowflake photo is of much higher quality than Frog's previous photos. One of the meanings of Bentley's snowflake photography is to open up the microphotography technology. Today, the microphotography technology has achieved the ability to distinguish atomic images, which has greatly promoted the development of modern science and technology.

Figure 4. American farmer and photographer Bentley is taking pictures of snowflakes.

Figure 5. Bentley's first snowflake photo successfully won the first snowflake photo, which made Bentley more fascinated by taking pictures of snowflakes. Bentley is often seen standing in the blizzard, picking up falling snowflakes with feathers or velvet, carefully placing the samples under a camera lens that is also outside. Bentley has obtained more than 5000 snowflake photos, and Bentley perfected the snowflake photography technique in the process. The second significance of Bentley's snowflake photos is to stimulate people's interest in snowflake research. More than 2500 pictures of snowflakes with lace designs are shown in his 1931 book Snow crystals, which fascinates people with hexagonal symmetries but different styles (figure 6).

Figure 6. Bentley's book Snow Crystal and the snowflakes he photographed in different shapes. Bentley noticed from his photography that although snowflakes are generally hexagonal, he has never taken two identical pictures-Every single snowflake is unique. The argument that each snowflake is different may not convince everyone. After all, the number of snowflakes being photographed is limited, and the definition of "different" is vague. However, it is inconceivable that snowflakes can show so many different forms that are known while maintaining hexagonal symmetry. "under the microscope, I found that snowflakes were amazingly beautiful, and it would be a pity if the beauty could not be seen or shared with others," Bentley wrote emotionally. "every crystal is an outstanding design, and none of it is repeated. Once the snowflake melts, the design disappears forever." Imagine how many snowflakes have fallen on the earth, but only a few have been recorded, which is a pity.

In order to give you a more intuitive understanding of the beauty and variety of snowflakes, you might as well add a few more snowflake photos obtained by modern photography (figure 7). If you think it's not enough, please use Snowflake, snowflake, snow crystal and other words to search.

Figure 7. Snowflake photos taken with modern technology. Now let's see what snowflake means. Snowflakes are centrally symmetrical and are always made up of six identical, left and right symmetrical branches. In scientific terms, snowflakes have D6 symmetry. However, this salient feature may be sufficient for describing microcrystals of materials such as ZnO-it is hardly fancy, but it is far from enough to describe snowflakes. Snowflakes can always change new patterns but maintain an approximate D6 symmetry as a whole. Why? Leaving aside the substantial problem that snowflakes are water solids, even if you construct snowflake patterns from a mathematical point of view, that is, to construct different patterns that maintain D6 symmetry, you will find that people's imagination is also very limited. Compared with the reality in nature, people's imagination is too pale.

The formation process and morphology of snowflakes are still a frontier topic that puzzles scientists today. With clearer and beautiful pictures of snowflakes, I thought we could understand the formation of snowflakes more deeply, but as a result, we became more confused about the atomic processes and thermodynamics of snowflake growth. It has now been determined that snowflakes have a generally consistent unique shape in different regions of the plane made up of temperature and water vapor supersaturation (figure 8), but show different shapes in different regions at a distance. For water to freeze, some water molecules first form some micron-sized ice nuclei, that is, a pre-nucleation process is required. The formation of snowflakes should have two processes: cold nucleation and growth of liquid droplets, and the shape of snowflakes can be classified as dendrite. The so-called Structure-dependent attachment kinetics (structure-dependent attachment kinematics) model about the mechanism of snowflake growth is only an improvement on the previous kinematic model of crystal growth, which is far from enough to answer the question of snowflake morphology.

Figure 8. The snow crystal morphology phase diagram in the temperature-water vapor supersaturation plane understands the shape of the snowflake. The author thinks that the key point lies in the nucleation process rather than the later dendrite growth process. It should be a phase transition process of supercooled water droplets large enough (and a process of volume expansion) rather than a growth process starting from scratch. To this end, there should be clear answers to the following three questions: 1. Why are three-dimensional droplets flaky when they become solid? two。 Why is the chip hexagonal; 3. How to construct such varied and different shapes while maintaining the symmetry of D6? Of these three questions, the second is probably the best one to answer. The configuration of hexagons can be found everywhere in nature, and hexagons are most suitable as elements (motif) for the requirement of a full plane (figure 9), because its topological charge, that is, the value of its V-E+F (V, number of vertices; E, number of edges; F, number of faces) defined by the author, is always zero. Of course, this fact does not require that water droplets become hexagonal symmetrical wafers.

Figure 9. Beehive. Hexagonal plaid arrangement is nature's favorite. After talking for a long time, there is no convincing answer to why snowflakes are so charming, but each snowflake is unique. Don't blame scientists, scientists really understand very few problems-scientists themselves are in a hurry. Finally, as a consolation, I will give you a hint to shoot snowflakes. What I fear most about shooting snowflakes is that they melt before they are well photographed. In order to capture beautiful snowflakes, choose items such as sweaters and silk fabrics that have poor thermal conductivity and are cool enough to accept snowflakes. Shoot outside in cold air and take macro shots with several magnification. Of course, the melting snowflakes are also beautiful (figure 10). As an inverse problem, perhaps the melting process of snowflakes will give us some inspiration about the mechanism of snowflake formation.

Figure 10. Notes on snowflakes starting to melt

Cao Zexian, Yi Niu extraordinary, Foreign language Teaching and Research Press (2016).

Philip ball,On the six-cornered snowflake, Nature 480,455 (2011).

[3] Kenneth G. Libbrecht, The physics of snow crystals, Reports on Progress in Physics 68855 (2005).

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