Colloidal crystals full of invisible vacancies

Daan Frenkel1 Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom

ll solids at finite temperatures contain point defects such as A vacancies and interstitials. Even at very low concentrations, such defects have a pronounced effect on the transport properties (e.g., electronic) of otherwise pure crystals. However, the ef- fect of point defects on the thermody- namic properties of crystals is small, simply because the mole-fraction of defects is low—at least, in one-component crystals. In fact, in most simple metallic and ionic crystals, but also in colloidal hard-sphere crystals, the vacancy fraction (i.e., the fraction of empty lattice sites) close to the melting transition is typically

substantially less than one part in a thou- 3 Fig. 1. Left: Snapshot showing the (projected) center-of-mass positions of a system of 40 hard cubes. sand. The article by Smallenburg et al. (1) Note that although the system contains 403 particles, there are 413 lattice sites, corresponding to a va- in PNAS presents an astounding simula- cancy concentration of more than 7%. This particular structure is mechanically stable, but the maximal tion result: some colloidal crystals may equilibrium concentration of vacancies is approximately 6% (1). Right: Snapshot of a system of hard have vacancy concentrations that are al- cubes. Figure shows a single plane that contains several extended vacancies in the plane. Some of the most three orders of magnitude higher extended vacancies have been highlighted in yellow. than what has been observed before in simple, one-component crystals. without “interdigitation”: that means that L over a single lattice spacing would cor- Smallenburg et al. (1) report numerical entire rows of particles can slide with re- respond to the displacement of a particle simulations of the phase behavior of sys- spect to each other without necessarily over distance L. Because mass transport in tems of hard, cube-shaped particles. At creating hard core overlaps. However, as fi a hard-cube crystal is relatively easy, such suf ciently high densities, such hard cubes Fig. 1 (Left) shows, the sliding of entire materials should be able to adjust their form a simple cubic crystal. The surprising rows does not happen—and this is proba- shape to that of the container walls. This result of ref. 1 is that these crystals contain bly due to the very existence of delocalized does not imply that the material is a liquid, an equilibrium concentration of vacancies vacancies. A vacancy can delocalize until nor even a supersolid—but such crystals that can be as high as 6%. This implies that it is blocked by a shifted row or by another could creep through holes. any particle will on average have more delocalized vacancy. The entropic cost Clearly, the work of ref. 1 raises many than one vacancy among its 26 neighbors. of delocalizing an entire row would interesting questions, such as, what hap- Using free-energy calculations, the authors therefore be very high owing to “vacancy pens for other particles that can form show that the very high vacancy concen- ” excluded-volume effects: delocalized va- space-filling structures? In fact, in a recent tration has a pronounced effect on the “ ” cancies behave as a polydisperse gas article, Agarwal and Escobedo (2) have location of the melting transition. of perpendicular hard rods (similar to the presented a beautiful overview of the Another striking feature of the calcu- well-known Zwanzig model for nema- phase behavior of a wide range of space- lations of ref. 1 is that the crystallinity of togens). One interesting question is: what filling, polyhedral particles. This article the samples with vacancies is actually determines the length of delocalized va- shows evidence for high diffusivity of hard higher than that of a corresponding system cancy? An upper limit is set by the va- cubes in the density regime where ref. 1 in which the number of particles is equal – cancy vacancy interaction. If we assume finds a high vacancy concentration. Agar- to the number of lattice sites. Looking at that line vacancies cannot intersect (and fi the average positions of the particles in wal and Escobedo (2) also nd high dif- that they are not orientationally ordered), fusivity for other polyhedral hard particles the crystal, one observes a perfect simple the maximum vacancy length L is de- systems in which the possibility for “row cubic lattice. However, looking at the in- termined by the vacancy density ρ, such translation” exists. However, the article stantaneous snapshots of the crystal (Fig. that ρL2 = O(1)—but, of course, the va- does not report collective diffusion con- 1, Left), one cannot “see” vacancies: the cancy length could be less. stants. Refs. 1 and 2 differ in their in- vacancies are delocalized along rows in the The high concentration of vacancies crystal lattice (Fig. 1, Right). This means also has a pronounced effect on the dif- terpretation of the phase that ref. 1 calls “ simple cubic and that ref. 2 calls cubatic. that one cannot really speak of vacan- fusivity in the crystal phase, because va- 3 cies” or “interstitials,” only of the differ- cancies facilitate particle diffusion. Ref. 1 shows that, at least for a 40 system, ence between the number of particles and translational order extends throughout the However, the fact that vacancies are — the number of lattice sites. The fact that delocalized should lead to a very in- system and this is why a discussion in the “point” defects are actually extended teresting consequence: mass transport will raises many interesting questions and be determined by a collective diffusion possibilities. The first is, of course: why are coefficient that should be significantly Author contributions: D.F. wrote the paper. vacancies delocalized? The most plausible higher than the single-particle diffusion The author declares no conflict of interest. explanation is related to the fact that coefficient. This is easy to understand: the See companion article on page 17886. cubes can form a space-filling lattice motion of a delocalized vacancy of length 1E-mail: [email protected].

17728–17729 | PNAS | October 30, 2012 | vol. 109 | no. 44 www.pnas.org/cgi/doi/10.1073/pnas.1215398109 Downloaded by guest on September 30, 2021 COMMENTARY

terms of a simple cubic crystal is justified. sequently extended by Donev et al. (4) to Finally, it should be stressed that crys- To know what happens in the thermody- systems of hard rectangles. Ref. 3 did not tals of hard cubes or other polyhedral namic limit, a full-scale finite-size scaling explore the presence of vacancies—and particles are not purely hypothetical sys- analysis would be required. Evidence for yet vacancies are likely to be just as im- tems: with the recent advances in the a bond-ordered liquid phase (a “tetratic” portant in two dimensions as in three. manufacture of tailor-made colloidal par- phase) in a system of 2D hard squares was Moreover vacancies make the system ticles, all of the phenomena discussed presented some time ago by Wojciechow- highly compressible, and this may facilitate in the present commentary can be ski and Frenkel (3). This work was sub- the formation of bond-ordered phases. investigated experimentally.

1. Smallenburg F, Filion L, Marechal M, Dijkstra M (2012) 3. Wojciechowski KW, Frenkel D (2004) Tetratic phase in 4. Donev A, et al. (2006) Tetratic order in the phase Vacancy-stabilized crystalline order in hard cubes. Proc the planar hard square system? Comp Meth Sci Tech 10: behavior of a hard-rectangle system. Phys Rev B 73: Natl Acad Sci USA 109:17886–17890. 2235–2255. 054109. 2. Agarwal U, Escobedo FA (2011) Mesophase behaviour of polyhedral particles. Nat Mater 10(3):230–235.

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