All the frost shapes derive from the underlying hexagonal structure of ice. How it appears depends among other things on humidity, the air temperature, whether the sky is clear or cloudy, the texture and conductivity of the surface and so on. In other words frost can be incredibly varied. Fern frost, so named for its leafy fern like swathes, is best seen after a cold night on old single-glazed window panes and on other smooth glassy surfaces like car bodywork.
Fern frost formation starts when the surface cools below that of the surrounding temperature by radiating its heat out to the empty night sky. Water molecules from the air condense onto rough (at an atomic level) nuclei – scratches, pits, dust particles. Once a tiny ice crystal has formed it spreads across the cold surface. Crystal growth is fastest at edges rather than on the large smooth planes of facets. We see large single crystal shapes when the initial growth nuclei are far apart.
The growth is dendritic, treelike, along the hexagonal crystal directions to give the characteristic ribbed and branched appearance. The complex but overall symmetric shapes of snowflakes are similarly produced. Why do the crystal intersections have sweeping gentle curves to give the fern leaf shapes? I don’t know. But whatever controls the overall shapes is a general effect.
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