Quasicrystalline order in self-assembled binary nanoparticle superlattices
Original entry: Sujit S. Datta, APPHY 225, Fall 2009.
D. V. Talapin, E. V. Schevchenko, M. I. Bodnarchuk, X. Ye, J. Chen, and C. B. Murray, Nature 461, 964 (2009).
quasicrystal, self-assembly, packing
Equilibrium phase transformations are ubiquitous in nature. Because of their complexity, it is often useful to focus on the 'simple' case of hard spheres, whose equilibrium phase diagram is dictated purely on entropic grounds. For monodisperse Brownian spheres with purely hard-sphere interactions, volume fraction is the only control parameter: for small volume fractions, the system is fluid-like; for intermediate volume fractions (between ~49% and 55%), a fluid and crystal phase coexist; for volume fractions larger than ~55%, up to the maximum FCC packing fraction of ~74%, the system is crystalline. For binary systems consisting of particles of two different sizes, the equilibrium phase diagram is much richer (e.g. see "Entropy-driven formation of a superlattice in a hard-sphere binary mixture" by Daan Frenkel's group in 2003), with three control parameters now - the volume fraction of each size particle, and the relative size difference between them. Further complexity develops in the equilibrium phase diagram with deviations from hard sphere behavior, such as the incorporation of van der Waals, Coulombic, and dipolar interactions. Different crystalline phases of nanoparticle systems have been experimentally observed (and theoretically predicted) as a result. This work is the first to demonstrate an equilibrium quasicrystalline phase, with clues suggesting that the phase is solely a general result of entropy, not the interparticle interactions.
The nanoparticles used in this study are made of two different materials, and have two different sizes, respectively (with size ratio ~2.7:1). Because of their surface chemistry, they do not aggregate due to van der Waals interactions, but have additional short range steric repulsions. Thus, the key control parameter in this system is the relative composition of the different nanoparticle components. Consistent with previous work, Talapin et al. found that "normal" crystalline superlattices of different structures form when one component dominates the composition of the system. These themselves have intriguing structures, resembling regular 2D Archimedean tilings of squares and triangles. More surprisingly, Talapin et al. found that for intermediate composition ranges, superlattices with quasicrystalline order formed -- specifically, the superlattices consisting of square and triangle packing motifs organized into a dodecagonal quasicrystal (possessing 12-fold rotational symmetry, and no translational symmetry). The three-dimensional material then consists of periodic stacks of these quasicrystalline layers, making their structure straightforward to analyze using transmission electron microscopy, as is done in this work.
Another clue towards understanding the formation mechanism of the DDQC phase is the observation that such phases tended to coexist with "normal" crystalline superlattices, with a sharp interface between the two. This is possible because the fundamental units of both phases are the same: either square or triangular tiles.
Intriguingly, nanoparticles of slightly smaller sizes or different materials (with a similar size ratio ~2.7:1 or ~3:1, respectively) also assembled into the dodecagonal quasicrystal (DDQC) phase, often extending up to thousands of particles on a side. This suggests that interparticle interactions (which would change for different materials and different particle sizes) do not play a significant role in the formation of the DDQC phase -- furthermore, the possible dependence on particle size ratio suggests that the formation of the DDQC phase may, in fact, be due to purely entropic considerations. However, it is important to note that whether or not the hard-sphere DDQC phase is a thermodynamically-stable phase is an open question, and may be strongly dependent on the manner in which the structures are formed (in this case, by slow evaporation of the suspending liquid). Further work exploring many different particle sizes, size ratios, compositions, and sample preparation mechanisms, as well as simulations of binary hard sphere systems, should help to elucidate the mechanisms driving the formation of this unique phase of matter.