The Wonderful Physics of Snowflakes

Biotop Christmas Calendar Day 6

This story is about a substance that couldn’t be any more contradictory: It is mundane in its omnipresence, while oozing with symbolism. It forms the basis of all life on this planet, while being capable of unleashing destructive forces of unrivalled power.

In spite of being formed from two of the three most common elements in the universe, pure, liquid water is an incredibly valuable good characterised by physical properties that are unusual to say the least. Water is a small molecule that occurs on earth in solid, liquid and gaseous form. This is remarkable, as most similarly small molecules would only occur in gaseous form under terrestrial conditions. Whether a substance occurs as solid, liquid or gas, is largely determined by temperature and pressure. If temperatures on Earth drop below 0°C the rules of the interplay between water molecules change substantially: Whereas water in liquid form is best described as a dancing frenzy of water molecules that only form connected groups for a short time, the dancing abates as water solidifies, and its molecules form tight formations.

If we take a close look at a snowflake, we have to take note of a certain sixness to it. Though, to say they were six-sided would do them a grave injustice, for they have many more sides than that! To capture this essence of sixness, we use the term symmetry. In the case of the six-fold rotational symmetry of the snowflake, this simply states that after turning the snowflake one sixth of a whole rotation, it appears indistinguishable from the original configuration. The symmetry of a snowflake is visible to the naked eye, yet it manifests the interplay of individual water molecules that each measure less than a fifth of a billionth of a meter.

Depending on pressure and temperature, water molecules can gather in a variety of formations giving rise to no less than fifteen different types of ice. However, most of these types of ice only form under special pressures and temperatures. Snow and virtually all other forms of ice on earth are formed by so-called ice Ih, which is characterised by hexagonal formations of water molecules that collectively give rise to a crystal lattice.

As opposed to molecules of liquid water, where the angle between two hydrogen atoms is 105°, in ice, the water molecule becomes stretched to better accommodate the molecules in the crystal lattice, resulting in a slightly larger bonding angle of 109.5°. Therefore, water molecules in ice need more space than the same number of molecules in liquid water. This accounts for the peculiar expansion of water as it is cooled. This behaviour that is very uncommon in the molecular world, and it contributes to the fact that earth is such a hospitable planet for life. For, were it not for its lower density, ice would not float on water, bodies of water would freeze from bottom to top. However, ice does float and aquatic organisms can spend cold periods in liquid water.

It is not a question of the crystal lattice if freezing water turns into snowflakes or ice, but depends on the environmental conditions of freezing. Snowflakes form from tiny droplets of supercooled water (water, free from impurities, that can remain liquid way below 0°C). Water molecules in supercooled water in a sense forget to freeze. However, if they encounter a speck of dust or a grain of sand, things take a quick turn. The surface of the impurity helps the water molecules to get into their hexagonal formation and if the surrounding air is full of tiny droplets of supercooled water, the snowflake can continue to grow. Quintillions (one followed by eighteen zeroes or 1018 for short) of water molecules try to get into formation as quickly as possible. In the crystal lattice, a water molecule can take up one of six distinct orientations. As the freezing of supercooled water happens very quickly, and the molecules in the droplets don’t have access to every possible site, the molecules will occupy the most accessible site in the lattice. This causes rays to form that grow faster along their tips. As the orientation and position of water molecules is random prior to solidifying, the appearance of shapes along the hexagonal lattice is governed by randomness, and small initial differences may give rise to large differences later on in the growth process. As the snowflake begins to fall towards the ground, the atmospheric conditions around it change Each snowflake experiences a subtly different journey.

All of these factors cause the snowflake itself to become a little symphony about its own existence, carved by pressure and temperature into its crystal lattice. Hence, it is little wonder that one commonly hears that each snowflake is unique! For compared to the number of theoretically possible snowflakes (a number that hard to put in words and even harder to write down, as it has a quintillion digits and is larger than the number of atoms in the universe), makes even the number of snowflakes that fall each year - roughly one septillion (1024) appear tiny. Still, the common wisdom about snowflakes is to be taken with a grain of salt, as the number of possible snowflakes is considerably smaller for smaller snowflakes, thereby rendering it more likely for two to be alike.

Even this slight rebuff of a romantic idea associated with snowflakes does little to break their spell. The sheer variety of small, differently sized surfaces in a manifold of orientations imbues snowflakes with that soothing capacity to drench a freshly snowed upon landscape in silence - a common theme among Christmas carols. It is also this very multifacetedness that makes the millions of snowflakes that form a little heap of snow appear completely white, in spite of them each individually being colourless. Snowflakes and ice fromed from compacted snow reflect white light almost entirely and thereby influence the capacity of our planet to reflect sunlight. As part of this so-called albedo, they are not only an expression of global climate, but play an important role in determining it. The melting of the pole caps reduces the capacity of the earth to reflect incoming sunlight and thereby its ability to stay cool…

This tale of the grain of dust that turned into the epitome of purity is as rich in facets as the snowflake itself. It is an epic journey that takes us from dancing particles so small that even the best microscopes cannot see them to processes of global scale. Yet it ends in a bittersweet tone: for all the beauty it encapsulates, the story of the common snowflake in all its complexity and fragility also serves as cenotaph of a looming climate catastrophe.

* To estimate the number of possible snowflakes, we can imagine a crazy bike lock, that is equipped with a little number wheel for each of the quintillion water molecules. Every wheel has six numbers on it that represent the six different orientations a water molecule can have in the crystal lattice. All we have to do is to calculate the number of possible combinations of this crazy bike lock. The number of combinations increases six fold with every wheel that is considered. In our case it increases six fold one quintillion times.

** Roughly a million billion kilogram of snow fall per year. An approximate weight of a billionth kilogram per snowflake allows us to estimate the number of snowflakes that fall every year to be on the order of one septillion (1024).

Further Reading:

  • Philip Ball - H2O A Biography of Water, 5.10.2000, ISBN: 9780753810927
  • Kenneth G. Libbrecht - snowcrystals.com The definitive resource by one of the world’s leading experts on snowflakes.
  • Ryo Kobayashi “Modeling and numerical simulations of dendritic crystal growth.” Physica D: Nonlinear Phenomena 63.3-4 (1993): 410-423. - Paper providing the theoretical basis for the simulations of snowflake growth.
  • Roberto Maffulli - iSeeing - Github repository with python library to produce simulations.

About the authors

Lukas Hutter studied chemistry in Graz and Systems Biology at the University of Oxford. He is a co-founder of Biotop and works as a teacher in Villach.

Roberto Maffulli is a specialist in fluid dynamics and currently works as a lecturer at Balliol College, University of Oxford.