Nucleation is the earliest stage of crystallization, in which atoms or molecules dispersed in a crystallization medium first come together to form ordered clusters known as nuclei. Crystal nucleation underpins a vast range of phenomena, from the solidification of rocks from molten magma to the hardening of biological tissues through the formation of various minerals, and the protein fibrillation and crystallization associated with a plethora of diseases. In many instances, nucleus formation represents the rate-limiting stage of crystallization and determines the main properties of a crystal population, including the type, number and size distribution of the crystals that form. In a paper in Nature, Zhou et al.1 report features of crystal nucleation that not only clash with several assumptions of classical nucleation theory, but also go beyond more recent non-classical models.

Read the paper: Observing crystal nucleation in four dimensions using atomic electron tomography

Crystal nucleation occurs in a medium (a solution, melt or vapour) that is supersaturated with respect to a crystal. In other words, the concentration of the material dissolved in the medium is higher than that at which the medium can maintain an equilibrium with the crystal phase of the dissolved material. However, nucleation has to overcome an energy barrier owing to the formation of new crystal surfaces in the ‘old’ supersaturated phase2. The nucleation barrier can be overcome by means of fluctuations that bring the local concentration and structure of the dissolved material close to those of the developing crystal2.

These basic tenets of nucleation, together with a kinetic model developed in the 1930s and 1940s3–5, constitute what is now known as classical nucleation theory (CNT). The main assumptions underpinning CNT are that a sharp interface separates the old and the new phases (Fig. 1a), and that the nucleation barrier is proportional to the surface area of the nucleus, with a coefficient of proportionality that is independent of nucleus size. Numerous systems have been found to follow CNT faithfully, mostly at relatively small supersaturations6,7.

Figure 1 | Models of crystal nucleation. a, The initial stages of crystallization involve the formation of nuclei — clusters of atoms or molecules that then grow into crystals. In classical nucleation theory (CNT), nuclei form at a particular shape and size, with sharp edges. Their constituents have the exact arrangement of the final crystal. Disordered constituents, pale blue; ordered constituents, dark blue. b, Some non-classical models propose that the constituents initially form a completely disordered precursor that is more concentrated than the starting phase, within which a nucleus emerges in a separate step. c, Other non-classical theories account for situations in which just one atom (or very few atoms) acts as a nucleus. d, A third category of non-classical theory suggests that nuclei adopt the lattice of the emerging crystal, but not the ideal ‘equilibrium’ shape expected in CNT. e, Zhou et al.1 report observations of nuclei that have diffuse edges and adopt non-equilibrium shapes that vary with time.

But in the past 15 years, elaborate experiments have accumulated evidence that many nucleation processes can behave quite differently from what is assumed by CNT. The discrepancies are particularly dire for crystal nucleation from dilute media, such as solutions and vapours. The deviant behaviours can be categorized into three groups. The first is two-step nucleation (Fig. 1b), in which nucleation is preceded and facilitated by the formation of disordered precursors, mostly dense liquids8–10. This model refutes the classical assumption that fluctuations of concentration in the old phase coordinate with structure fluctuations to produce an ordered crystal nucleus11.

The second category is barrier-free nucleation (Fig. 1c), in which the nucleation barrier drops below the kinetic energy carried by atoms or molecules at a given temperature, even at moderate supersaturations, as the nucleus size shrinks to one atom or molecule12. Tiny nuclei are too small to produce a defined interface with the surrounding medium, and therefore require special theoretical treatments to describe their formation13. The third category is for nuclei that have adopted the structure of the emerging crystal phase, but have shapes that do not minimize the nucleation barrier (Fig. 1d); such ‘non-equilibrium’ nucleus shapes can form if the kinetics of the processes that restructure the emerging new phase are slow14,15.

Studying nucleation processes is a challenge, because it is devilishly difficult to image the structure of nuclei. Methods that have sufficiently high resolution to detect individual atoms or molecules have minuscule areas of view, and therefore struggle to find nuclei — which form in extremely small numbers and constantly move when in low-viscosity environments. Zhou et al. overcame this problem by studying crystal nucleation in nanoparticles of an iron–platinum alloy.

The atoms in the alloy have a disordered structure that, when heated, undergoes a solid-to-solid transition that can produce a tetragonal lattice. The tetragonal nuclei that emerge during this transition are immobile, and can therefore be studied using an imaging method called atomic electron tomography (AET), which can visualize all the atoms in a nanoparticle. The authors used AET to trace the positions of individual metal atoms as they lined up into crystal nuclei in the iron–platinum alloy. They appropriately describe this amazing experimental capability as 4D atomic resolution.

Zhou and colleagues noticed three nucleation behaviours (Fig. 1e) that go beyond both CNT and current non-classical models. First, they observed that the order parameter of a nucleus, which in this case quantifies how closely the atoms in the nucleus adopt a tetragonal arrangement, is not uniform. Instead, it varies with the distance from the nucleus core (which consists of one to a few atoms that have the maximum order parameter). As a result, the interface between the nucleus and the surrounding phase is not sharp, but diffuse, and its structure varies in time as the size of the nucleus increases. Second, the researchers observed that the overall shape of the nucleus is neither spherical nor tetragonal — the two shapes that would minimize the energy of the nucleus, assuming that the energy is proportional to the nucleus surface area (as is assumed in CNT). And third, the nucleus size and shape fluctuate broadly, which Zhou and co-workers interpret as evidence that the critical size of the nucleus (the size that has the highest energy) is not fixed, but varies in line with the order parameter.

Researchers have previously generally assumed that diffuse interfaces can form16, albeit with some different features from those actually observed by Zhou and colleagues. However, the lack of methods that have sufficiently high spatial and temporal resolution to monitor crystal nucleation in real time has led to ad hoc assumptions about the structures of interfaces, and therefore also about their consequences for the nucleation barrier and the nucleation rate.

A phase transformation in a metal alloy is a convenient model system for the first application of 4D AET. However, the burning problems in the field of nucleation relate to the formation of biological minerals and disease-associated aggregates of proteins and small molecules, the production of pharmaceuticals and fine chemicals, and to other processes in which materials are synthesized in solution. A desirable future development would therefore be for AET to be implemented in liquids, used in combination with methods that could constrain the positions of emerging nuclei (for example, by using nanometre-scale pores, or laser systems known as optical traps). Zhou and colleagues’ findings will also spur the development of a general theory that accounts for diffuse and dynamic interfaces, and could thus predict the magnitude of nucleation barriers and the rate of formation of crystal nuclei.

Leo Tolstoy’s novel Anna Karenina begins with the immortal words “All happy families are alike; each unhappy family is unhappy in its own way”. The main message of Zhou and colleagues’ paper can be summarized in a similar way: all nuclei that adopt an equilibrium shape are alike; every non-equilibrium-structured nucleus has its own shape. Moreover, the researchers demonstrate that non-equilibrium nuclei shapes not only are diverse, but also vary in time, and therefore probably enforce disparate nucleation pathways.