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Solid–solid phase transitions can significantly change the physical properties of crystalline solids, such as

metals, alloys and ceramics. For example, the metallic and ductile α-tin phase can deteriorate to the non-metallic and brittle β-tin, and the transition between a tetragonal to a monoclinic lattice in zirconia ceramics can cause Zr-based implants to fail after a few years1. Yet the molecular or atomistic mechanisms by which solid–solid transitions take place remain to be revealed. One main reason for this is that detecting molecular systems with sufficient spatial and temporal resolution to capture the atomistic events leading to a first-order phase transition is extremely difficult. However, this problem can be circumvented by using model colloidal systems instead, as colloidal particles are of the order of a few nanometres to micrometres and diffuse sufficiently slowly to be accurately tracked with a microscope. Using an experimental model for hard colloids, Yilong Han, Arjun Yodh and colleagues now show in Nature Materials that a phase transition between two colloidal crystals of the same chemical composition but different structure occurs through a transient liquid intermediate2.

Using single-particle-resolution video microscopy, Han and co-authors observed that the transition from a square lattice to a triangular lattice in colloidal films of hard spheres proceeded via two steps. First, a liquid nucleus formed in the metastable ‘parent’ crystal (the square lattice), and then the triangular lattice (the thermodynamically stable crystal) grew within the liquid nucleus (Fig. 1a). Why is this a more favourable pathway than a simpler one-step mechanism, that is, the formation of a nucleus of the thermodynamically stable crystal phase in the parent phase? Because the interfacial free energy between the crystals is larger than that between the square lattice and the liquid, the authors argue that the formation of an intermediate liquid nucleus is preferable, as it prevents the formation of crystal/crystal interfaces. This is reflected in the nucleation free-energy profiles: whereas the one-step pathway shows a single maximum, which corresponds to the critical

nucleus of the stable crystal, the two-step pathway features two maxima, these corresponding to the nucleation of a liquid nucleus in the parent metastable crystal and to the nucleation of the stable crystal within the liquid nucleus (Fig. 1b). The lower free-energy penalty for the two-step pathway makes this the preferential route. In fact, this is an example of the well-known Ostwald’s step rule, which states that transitions from one phase to another occur through the lowest free-energy pathway3.

Although two-step pathways involving an intermediate nucleus have been identified before (for example, in protein crystallization and in the freezing of hard spheres, which are believed to occur via precursors such as dense4,5 or pre-ordered clusters6,7), the unexpected finding in the two-step solid–solid transition reported by Han and colleagues is that the intermediate nucleus between two solid phases is a liquid. This pathway shares common features with that of another solid–solid phase transition, that between an amorphous solid (or a glass) and a crystalline solid. Indeed, recent computer simulations have shown that when a colloidal glass devitrifies (that is, recrystallizes), the stable crystalline phase grows as a consequence of the

nucleation of fluid regions where particles move cooperatively for a short period8 (resembling avalanches). However, in this case the stable crystal phase does not grow inside the fluid nuclei but in neighbouring regions of incipient crystal order. Hence, the fluid avalanches do not lower the free energy between the solid phases but ‘shake’ the nearly ordered regions so as to induce their crystallization.

The two-step nucleation mechanism demonstrated by Han and co-workers could be applicable to a broader class of systems. In fact, there is previous indirect evidence that the transformation between graphite and diamond also goes through an intermediate fluidic state9. Yet two thermodynamic requisites are needed for the mechanism to occur: first, the liquid–solid interfacial free energy has to be lower than that between the solids. This is likely to be the case for many systems (still, one should keep in mind that the interfacial free energy involving a solid phase cannot yet be measured accurately, as exemplified by the wide range of values reported for the ice/water interface10). Second, the liquid must have a thermodynamic driving force to nucleate in the parent solid — that is, the chemical potential of the liquid must be lower than

CRYSTAL–CRYSTAL TRANSITIONS

Mediated by a liquidThe nucleation of a crystal within another can involve intermediate liquid nuclei.

Eduardo Sanz and Chantal Valeriani

Intermediate state(metastable liquid)

State

Free

ene

rgy

Initialstate

Finalstate

Temperature

LiquidSolid A

Pres

sure

Solid B

a b c

Figure 1 | Two-step nucleation pathway in the transformation from a square lattice into a triangular lattice in thin films of hard colloidal spheres. a, Single-particle-resolution microscopy image (the inset shows a schematic) of the nucleation of a triangular lattice (blue) within a liquid nucleus (red) formed in the ‘parent’ square lattice (green). Scale bar, 5 μm. b, The free-energy profile of a two-step pathway involving liquid nuclei is lower than the direct nucleation of one crystal within the other (arrows indicate the direction of crossing of the free-energy barriers to the formation of interfaces). c, Schematic phase diagram of a system with two solid polymorphs and a liquid. For solid A to transform into solid B via an intermediate liquid nucleus, solid A must be quenched in the stability region of solid B at temperatures where the liquid is more stable than solid A (that is, the green region on the right-hand side of the dashed line). Panels a,b reproduced from ref. 2, 2015 Nature Publishing Group.

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that of the parent solid). This condition may not be satisfied generally because the chemical potential of the liquid depends on thermodynamic conditions (Fig. 1c). If the liquid is less stable than the parent solid (that is, if it has a higher chemical potential), the solid–solid transition would most likely occur through a diffusionless martensitic route.

Experiments and computer simulations of model colloidal particles are ideal for investigating whether solid–solid transition pathways depend on thermodynamic conditions, and can also provide clues on phase-transition mechanisms in atomic or molecular systems11,12. Still, accurate knowledge of the equilibrium phase diagram

is required, which is difficult to achieve experimentally. In this respect, simulations can provide useful guidance13 — in fact, the stability range for the triangular and square lattices had been predicted earlier14. With advances in particle synthesis, video microscopy and the characterization of model colloidal systems, one would expect that the effects of constant pressure, particle asymmetry and particle softness — conditions that are generally more realistic — will soon be evaluated. ❐

Eduardo Sanz and Chantal Valeriani are in the Departmento de Química-física I, Universidad Complutense de Madrid, 28040 Madrid, Spain. e-mail: esa01@quim.ucm.es

References1. Shen, J. Z. & Kosmač, T. (eds) Advanced Ceramics for Dentistry

(Elsevier, 2014).2. Peng, Y. et al. Nature Mater. 14, 101–108 (2015).3. Ostwald, W. Z. Physik. Chem. 22, 289–330 (1897).4. ten Wolde, P. R. & Frenkel, D. Science 277, 1975–1978 (1997).5. Schilling, T., Schope, H. J., Oettel, M., Opletal, G. & Snook, I.

Phys. Rev. Lett. 105, 025701 (2010).6. Kawasaki, T. & Tanaka, H. Proc. Natl Acad. Sci. USA

107, 14036–14041 (2010).7. Tan, P., Xu, N. & Xu, L. Nature Phys. 10, 73–79 (2014).8. Sanz, E. et al. Proc. Natl Acad. Sci. USA 111, 75–80 (2014).9. Shekar, N. C. & Rajan, K. G. Bull. Mater. Sci.

24, 1–21 (2001).10. Prupacher, H. R. J. Atmosph. Sci. 52, 1924–1933 (1995).11. Khaliullin, R. Z., Eshet, H., Kühne, T. D., Behler, J. &

Parrinello, M. Nature Mater. 10, 693–697 (2011).12. Sanz, E. et al. J. Am. Chem. Soc. 135, 15008–15017 (2013).13. Vega, C., Sanz, E., Abascal, J. L. F. & Noya, E. G.

J. Phys. Condens. Matter 20, 153101 (2008).14. Fortini, A. & Dijkstra, M. J. Phys. Condens. Matter

18, L371–L378 (2006).

MATERIA

L WITN

ESS

Origami — the folding of flat sheets — has been a well-advertised approach to engineering structures with diverse properties that can be efficiently and easily collapsed and unfolded1. But there is another constructive paper art in Far East Asia that has received less attention: kirigami, in which paper is cut into intricate patterns. This approach often involves folding too: the archetypal kirigami form is a scene, often a building, made from strips and facets that emerge from the hinge of a folded sheet.

Although kirigami is essentially of Japanese origin, it has long been familiar throughout the world in the form of the snowflake decoration made by folding and nicking a sheet (a method sadly apt to yield snowflakes with four-fold rather than six-fold symmetry). Paper-cut snowflakes — simply outlines rather than genuine kirigami — featured in one of the first serious studies of their forms, Cloud Crystals (1864) by the keen-eyed American amateur Frances Chickering.

Kirigami has, however, so far enjoyed scant attention in materials science — among rare examples, there has been interest in applying its methods at the nanoscale to graphene to alter its mechanical properties2,3. Now, Cho et al. have attempted to give kirigami engineering a more systematic conceptual basis4. They describe a general approach for

enabling a sheet of flexible material to adopt an arbitrary geometry by reverse-engineering the system of cuts needed to attain it. No folding is employed here: the deformations of the sheet come solely from making a series of slits that fragment the sheet into smaller blocks connected at their corners, which will then rotate under tension so that the blocks open out into an expanded network.

In general these blocks might be squares or triangles. Their rotation depends on leaving a small amount of material at the ends of the cuts, and on these hinges being sufficiently flexible to deform while strong enough not to simply tear. The design parameters here are material-dependent: the researchers demonstrate their ideas with elastomeric sheets, but say that metals might be used if the hinge stresses can be kept below the yield strength.

The amount of expansion of the fabric (shown here to reach up to 800%) can be adjusted by imposing a fractal hierarchy on the cuts. Each square block, say, can be divided into smaller squares and so forth. By varying the degree of the hierarchy from place to place, the sheet can adopt all kinds of curved grid-like geometry, including non-Euclidean ones. As a demonstration, Cho et al. cut a silicone rubber sheet, coated with a layer of carbon nanotubes to make it electrically conducting, so that it

can be wrapped around a baseball without wrinkles while retaining the connectivity that enables it to wire up a light-emitting diode across the ball’s surface.

There is an echo here of the reverse problem of projecting the Earth onto a flat world map. Cho et al.’s method of covering a sphere with a sheet is certainly more elegant than the ungainly Goode homolosine cartographic projection, with its awkward rents in the oceans — but at the considerable cost, perhaps, of fragmenting the surface into countless little pieces. ❐

References1. Schenk, M. & Guest, S. D. in Origami 5: Fifth

International Meeting of Origami Science, Mathematics, and Education (eds Wang-Iverson, P., Lang, R. J. & Yim, M.) 293–305 (CRC Press, 2011).

2. Blees, M., Rose, P., Barnard, A., Roberts, S. & McEuen, P. L. Bull. Am. Phys. Soc. 59, abstr. L30.00011 (2014).

3. Qi, Z., Park, H. S. & Campbell, D. K. Preprint at http://arxiv.org/abs/1407.8113 (2014).

4. Cho, Y. et al. Proc. Natl Acad. Sci. USA http://dx.doi.org/ 10.1073/pnas.1417276111 (2014).

THE FINAL CUT

PHILIP BALL

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