Crystallization is a rare case of a phenomenon which is familiar in everyday life and yet is the focus of intensive research at the forefront of physics, chemistry, and materials science. Common examples of crystallization are the formation of snowflakes and ice crystals, crystallization of honey and other forms of sugar, and the classic example of the salt crystals left behind when water is evaporated from a briny solution. But the importance of the process is far larger: precipitation of solids from solution is a key process in chemistry, and many pharmaceuticals are sold in crystalline form. The list of applications that depend on crystal formation is endless.

Crystallization involves two separate processes. The first is “nucleation,” in which a small number of molecules come together to form clusters. Normally, small clusters are unstable and will quickly fall apart, but occasionally, by chance, one grows large enough to become stable. Such “post-critical” clusters will then grow until all available material has been used, and this second growth stage is a fundamentally different process.

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An enormous amount of research is performed on both processes, but nucleation remains more mysterious for several reasons. Most importantly, the clusters involved are usually very small – consisting of only tens or hundreds of molecules – and form rarely. This means that their formation happens infrequently, even given the unimaginable number of molecules in a solution in a test-tube, so it is impossible to know where to watch to see it happen. This means that it is hard to directly observe crystal nucleation, and researchers are forced to rely on indirect methods that only reveal partial pictures of what happens.

This image shows a crystal forming from a weak solution. The intensity of the color indicates the density of the crystal-forming species. At the center, the molecules are localized into a regular crystalline array and around the outside is a diffuse liquid-like layer. Image courtesy Jim Lutsko.

Nevertheless, scientists have studied crystallization since the 19th century, and by the mid-twentieth century, a set of tools called Classical Nucleation Theory (CNT) had been developed, which remains the primary paradigm used to understand nucleation in general and crystallization in particular(1). These tools are constantly being modified and improved, but the basic ideas have remained the same for a long time. Now, these ideas are being challenged by a revolution in our capability to observe events at the nanoscale(2) and by our ability to perform computer simulations of rare events like nucleation(3).

This research has shown that crystal nucleation is far more complex than previously believed. We now know that in many cases crystals do not form directly from a solution, but rather, they begin as small disorganized clusters – called precursors – that grow and eventually transform into crystals. There are many open questions about this process: are the initial clusters liquid-like or amorphous solids? Why are they able to live long enough to transform into crystals? Why is this the preferred pathway for forming crystals rather than directly creating crystals to start with? The tools of CNT have been used to address these questions, but they strain the framework.

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Recently, sophisticated theoretical tools from statistical physics and the mathematics of stochastic processes have been brought to bear on this problem in an attempt to go beyond CNT. The goal of this work is to start with nothing except the form of the molecular interaction forces and be able to predict the structure of the crystals that will form as well as the pathway of their formation such as the existence and nature of any precursors. Like CNT itself, this theoretical framework, called Mesoscopic Nucleation Theory or MeNT, is more general than just its application to crystallization: it applies equally, for example, to the nucleation of liquid droplets from an oversaturated vapor. And, importantly, CNT can be derived from MeNT so that the two are not different approaches – rather, MeNT is simply a more fundamental framework which, when combined with certain simplifying assumptions, reduces to CNT.

The extension of MeNT to crystallization has been described in a recent paper published in Science Advances(4). The proof-of-concept calculations presented in this work show indeed the formation of crystals consisting of several different phases. First is the formation of liquid-like droplets. These are not simple but consist of alternating spherical layers of high and low density “shells” – a phenomenon common in confined liquids but often neglected in simpler approaches (like CNT). These droplets can – rarely – grow via thermal fluctuations, and, as they do so, the shells become more distinct with the density in the dense shells increasing and the density in the region between them decreasing.

At a certain point, the molecules which are free to move around in the liquid state start to become localized into a solid-like structure at the core of the droplets. This is perhaps due to the density in the dense shells passing a stability threshold beyond which the formation of solids is inevitable. The resulting structures are thus solid-like in their core but surrounded by a liquid-like coating. The calculations show that these can be metastable, thus making them candidates for the precursors observed in experiments. Eventually, these grow large enough, with more and more of the interior of the cluster converted to crystal, until they become stable and nucleation gives way to crystal growth.

The accompanying gif shows a sequence of snapshots or “images” taken along the nucleation pathway for the formation of a crystal. The intensity of the color is proportional to the log of the density. In liquid-like regions, the density (which is the same thing as the probability to find a molecule at a given point) is smeared-out as molecules are free to move around. IN solid-like regions, the molecules become localized near lattice sites leading to small areas of high density separated by interstitial regions of very low density. Image courtesy Jim Lutsko.

While no single set of calculations can address all of the questions researchers have concerning the formation of crystals, this work does offer a way forward that is promising. Complementing the remarkable advances made in recent years in experiments and simulations with sophisticated theoretical tools can only accelerate progress in understanding this fascinating phenomenon.

These findings are described in the article entitled How crystals form: A theory of nucleation pathways, recently published in the journal Science Advances.

References:

  1. D. Kashchiev Nucleation: Basic Theory with Applications, Oxford: Butterworth-Heinemann (2000).
  2. R. P. Sear, The non-classical nucleation of crystals: Microscopic mechanisms and applications to molecular crystals, ice and calcium carbonate. Int. Mater. Rev. 57, 328–356 (2012).
  3. G. C. Sosso, J. Chen, S. J. Cox, M. Fitzner, P. Pedevilla, A. Zen, A. Michaelides, Crystal nucleation in liquids: Open questions and future challenges in molecular dynamics simulations. Chem. Rev. 116, 7078–7116 (2016).
  4. J. F. Lutsko, How crystals form: A theory of nucleation pathways, Science Advances 5, eaav7399 (2019).

About The Author

My background is in nonequilibrium statistical mechanics and nowadays most of my time is spent on the study of crystallization and crystal growth. Previously, I have worked on non-classical diffusion, granular materials, symbolic learning, kinetic theory and the mechanical properties of solid interfaces. My work involves a combination of analytic theory, numerical analysis, and computer simulation and is currently funded by the European Space Agency, via the Belgian Science Policy Office, and the EU Horizon 2020 program, via the AMECRYS project.

Jim Lutsko is a research scientist at the Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles.