Crystallization is the process by which a solid forms from solution, where the atoms or molecules become highly organized and form a structure.

Some aspects of the process, particularly nucleation is still not completely understood, despite intensive research going back more than a century.

Crystallization has been called “one of the most secretive processes in nature”. The mystery lies in how structures with long-range order form from building blocks that only interact with their local neighbors. The most extreme case is when solids form from weak solutions, in which the molecules begin widely separated so that not even local structure is present.
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One of the most important insights from the recent work has been the discovery of nonclassical pathways for crystallization. The naive picture of crystallization is that a few molecules come together by chance and happen to be arranged in a crystalline form. Additional molecules attach one by one, gradually building a larger structure. Small clusters are unstable because most molecules are near the surface and do not have the correct number of neighbors, but large enough clusters are stable.

Source : How crystals form: A theory of nucleation pathways in Science Advances, Vol. 5, no. 4,

Beginning the 1960s, a deeper understanding of particle physics made it possible to start looking for an overall mathematical description of the crystallization process. But, despite decades of research, it still hasn't been fully established.

While crystallographers study the properties of some periodic arrangements and compare them, there remains a more fundamental question: why is it favorable (at low temperature) for the atoms to spontaneously arrange themselves on a periodic array? This periodic order seems to only appear in the limit of a large number of particles, which makes the question particularly difficult.

Source : The Crystallization Conjecture: A Review. EMS Surveys in Mathematical Sciences, 2015, 2 (2), pp.255-306.

This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on R^3N where N is the number of particles, which tends to infinity.

[source as above]

The paper cited describes the technical details of progress-so-far (as at 2015)

Also see : Cloud ice formationplugin-autotooltip__plain plugin-autotooltip_bigCloud ice formation

The formation of ice crystals have important implications for stratospheric ozone chemistry, cloud dynamics, rock weathering, and hydrate formation etc., however the exact mechanisms by which microscopic particulate matter 'seeds' ice-crystals are unknown.

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