### Snowflakes

#### by Willie Wong

It took me two tries to get out of my flat this morning. I really ought to get into the habit of *looking out the window* in the morning; too often do I open the front door, ready myself to step out, only to turn back to fetch my umbrella. The annoying thing about snow is that I can’t hear it, unlike the pitter-patter of rain.

Somehow or another I ended up looking at Wilson Bentley’s micro-photographs of snow crystals. And a question forms in my mind, “Why are they all so symmetrical?” If *all* snowflakes were to look alike, then perhaps the dynamics leading to the formation of snow crystal is stable, and the global convergence unto a completely symmetrical pattern would not be surprising. But not all snowflakes look alike. In fact, colloquially we speak of them as each completely different from every other. This implies that the dynamics of snow crystal growth should be at least somewhat sensitive to atmospheric conditions in a local scale (and perhaps to the nucleus that seeds the crystal) so that the seemingly random to-and-fro dance as the snowflake falls from the sky can effect different shapes and branches.

Now, much experimental evidence has gone to show that the formation of ice crystals tends to by catalyzed by impurities. Pure water can be supercooled, in normal pressure conditions, to temperatures below 273 Kelvin. But in these situations a single mite of impurity dropped into the water can cause the entire beaker to freeze over suddenly. Similarly, ice crystals in the upper atmosphere tend to form around impurities: bacterium floating in the air, dust or ash, or perhaps particles introduced artificially. So one may surmise that the fact that all 6 branches of a snowflake grows in the same way because, somehow, the eventual shape of the snowflake is already encoded in the original formation of the central nucleus. Let me try to explain why this hypothesis is not very convincing. I’ll make one a priori assumption, that the growth of a crystal structure is purely local, and not due to some long-range interaction.

To draw an analogy, consider a large group of acrobats. They are trying to bring themselves into a formation around a leader. Disallowing long-range interaction can be thought of requiring that the leader cannot shout out orders to individual troupe members. But we can allow passing of information by short-range interactions, i.e. whispering instructions to the people already in formation. So the leader stands alone at the start. Then he grabs on a few people nearby to form the nucleus. Then he tells each of the people he grabbed a set of instructions on how to grab more people, where to put them, and what instructions to pass on to those people (included in these instructions are instructions to be passed on to the even more remote layer of acrobats and so on). Then if the instructions were passed correctly, a completely ordered pattern will form. But as anyone who has played the game of telephone can testify, in these scenarios some errors will always work its way into the instructions. In the physical case of snowflakes, these are thermodynamical fluctuations. So some irregularities should happen. Now, if the instructions the leaders were trying to pass down were very short and easy to remember, the errors tend not to build up, and the formation will, for the most part, be correct. But keeping the message short has the drawback that the total number of formations one can form is fewer. In the snowflake case, one can imagine somehow each small group of molecules in the snow crystal can encode some fixed amount of information. If the encoding is very redundant (so the total number of shapes is small), then the thermodynamical fluctuations will not be likely to break the symmetries between the arms. But considering the large number of possible shapes of snowflakes, such encoding of information should be taxed to the limit, and small fluctuation (errors in the game of telephone) should be able to lead one arm to look drastically different from the others. One possible way to get around this difficulty would be to use some sort of self similarity principle. But this will suggest the snowflakes are true fractals, which they are not.

So it seems that we must accept that the variety of snowflake shapes comes not from a central command, but from local variations of the environment (humidity, temperature, pressure, etc.) when the snowflakes are forming. But in this case, why aren’t there asymmetrical snowflakes? A bit of web search shows that no, not all snowflakes are completely symmetrical, an answer that makes me somewhat happier. When something is *never* observed to happen, we tend to attribute it to some sort of universal law: gravity is always attractive, magnetic monopoles don’t exist, etc. But when something is observed to happen, just rather infrequently, then we can try to find some sort of statistical/physical justification for its rarity. In the case of snowflakes, the prevailing wisdom is that typically, on the scale of the size of a snowflake, the environmental variables do not carry too big a gradient. So the symmetry and complexity of the snowflakes can be explained this way

- Symmetry: the original nucleus carries a six-fold symmetry due to it being the preferred crystal structure of water. The growth of the crystal is deterministic and depends only on local conditions: the current (local) shape of the crystal and the ambient environment. Since the environmental variables do not carry a big gradient, we can assume that the environment variables are the same across all growing interfaces. Thus the determinism preserves the original symmetry for the crystal growth.
- Complexity: the complexity of the shapes is due to the environmental variables. While the growth is not sensitive to small changes, it is to big ones. When the snowflake is blown from one place to another by the air current, the environmental variables may undergo a large change. In other words, we can plot the environmental variables as functions of time. This functions are what determine how the crystal grows over time.

So far, however, we’ve neglected a very important question: if the initial crystal nucleus forms with a six-fold symmetry (which essentially means that the initial crystal nucleus forms a hexagon), why do the new layers not only grow as a large hexagonal plate, but instead form dendrites at times? As it turns out, the boundary layer dynamics is size dependent. When the crystal grows, it “sucks” water vapor out from the air surrounding it. The water vapor then is replenished by diffusion from the air even further away. And the cycle repeats. Now, suppose an ice crystal can only grow if the local humidity is at a certain level. After it sucks the vapor out, the humidity drops below the level, and we have to wait until replenishment. A process like this is what is called (locally) diffusion-rate limited. The faster the diffusion of water vapor, the faster the crystal can grow.

Now, for a hexagon, we see that on an edge, the water vapor can only be replenished by basically diffusion from straight out. On or near a corner, however, there are more “neighboring area” from which to draw replenishment. (Imagine the plane being divided up into a hexagonal grid like those used in tabletop games. Draw a large hexagon on the grid. For a grid-point on the boundary, there are two adjacent points outside the hexagon. But for a corner point, there are three adjacent ones.) For a sufficiently small hexagon, virtually every edge point is close enough to a corner so that the entire boundary layer is replenished at the same rate. (Going back to the grid picture, look at a small hexagon with two grid points to a side and one point in the center. Then every “edge” point is actually also a corner point.) For a larger hexagon, a difference can be seen between corners and edges. This suggests that then a crystal will start out like a hexagon, and grow as a hexagon for a while. Then when it gets to a certain size, the corners will start to grow noticeably faster, and a dendrite will form. The critical size is determined by the rate of diffusion of the water vapor, which is dependent on the overall environmental variables. (Going back to the grid picture, the distance between grid points should roughly be the rate of diffusion multiplied by the time it takes for layer of deposits to form. And the critical size will be when the size of the crystal occupies a region that has at more than two grid points on each other edge.) So morally speaking, on a very wet day we expect snowflakes with very large inner hexagonal plate. If the snowflake were grown in an environment where the environmental variables are held constant, then this type of growth will typically lead to a fractal, with the size of the environmental variables determining the characteristic shapes and sizes of the “fractal element”. Now add in fluctuations of the environmental variables over time, and we can roughly see how the interplay between the changing characteristic shapes and sizes leads to the great variety of snowflakes seen.

A lot of mathematics (especially of the numerical type) has gone into the study of growth of snow crystals. For an overview of the fundamentals, I suggest this article from the American Educator by Ken Libbrecht. To see some early work on diffusion-limited processes, this paper of Miyazima and Tanaka has some good references; and for some modern numerical simulations using a more sophisticated version of what I described above, the work of Gravner and Griffeath (esp. the second paper in that webpage) gives good discussions on both the theory and the numerical techniques.