Computational explorations in the space of one-component crystals

How complex an ordered assembly can be made using just a single particle type? When the particles are atoms, and the ordered assemblies are limited to periodic crystals, the range of possibilities has been comprehensively mapped out (1). But the particles need not be atoms. Colloids, nanoparticles, proteins, and DNA origami provide new building blocks whose shapes and interactions can be potentially fine-tuned to achieve a given assembly target. Furthermore, molecular simulations provide a means to systematically explore the space of possible particles mapping out the crystal forms that result. In PNAS, Dshemuchadse et al. (2) do just this for particles interacting with isotropic multiwell potentials, finding many hitherto unknown and surprisingly complex one-component crystal structures.

Fig. 1 illustrates some of the possible dimensions of particle complexity and the ordered structures that can result. At its origin is the simplest case, an isotropic particle interacting with a simple potential—the canonical examples being a hard sphere and a Lennard-Jones particle—that favors simple crystals with a high packing fraction, such as the face-centered-cubic (fcc) lattice. Atomic systems span the axes of directional bonding (as exemplified in covalent materials) and more complex isotropic interactions (including those with many-body character). Colloidal particles, however, provide access to particles of different shape, the two example axes illustrated here being those of increased faceting and extreme aspect ratios (3).

Fig. 1.

Example dimensions of particle complexity and the one-component ordered assemblies that can result.

The type of order that can result is not just limited to periodic crystals. Although quasicrystals have only been observed for multicomponent atomic systems, namely, in metallic alloys, simulations have, for example, shown that dodecagonal quasicrystals spontaneously form for hard tetrahedra (4). Particles with extreme aspect ratio naturally lead to liquid crystalline orderings. For example, simple cylindrical rods form nematic and smectic phases, and, as the particle complexity increases, further phases become possible. Chiral rods typically form cholesterics, and recent experiments have shown that bent rods can form splay−bend phases (5).

Molecular simulations provide an ideal means to explore the fundamental question of how particle properties determine the ordered forms those particles adopt. Although this used to be mainly done through specific examples, or mapping out the behavior as a function of one particle variable [e.g., aspect ratio for the liquid crystal phases formed by rods (6)], efficient simulation algorithms and high-performance computing mean that systematic surveys of particle space can now be achieved. The Glotzer group has been one of the pioneers of this approach. For example, in 2012, they surveyed the assembly behavior of 145 convex polyhedral particles, finding liquid crystals, plastic crystals, and quasicrystals, as well as orientationally ordered periodic crystals (7); furthermore, the novel behavior observed in this study has stimulated an intense further exploration of the ordering behavior of faceted particles.

Although there has been significant interest in interparticle potentials with multiple wells because of their potential for structural complexity (8–11), the study of Dshemuchadse et al. (2) provides a systematic survey of the crystal structures that can be formed. The authors study two slightly different potentials. Both are isotropic pair potentials that generally have two minima separated by a barrier as a function of the interparticle distance. Each potential has two parameters; these effectively control the relative positions and the relative depths of the two minima. To explore these parameter spaces, the authors perform thousands of large-scale molecular dynamics simulations, each corresponding to a different point on a finely spaced two-dimensional (2D) grid in these spaces. In each simulation, the system is slowly cooled with the aim of promoting the nucleation and growth of a single crystal of the preferred form. Longer simulations are performed at those grid points where the ordering was initially unclear. Such a survey represents a prodigious computational and analysis task.

The result is two 2D crystal structure maps that reveal islands of stability for 31 different crystal structures, 16 of which have previously been observed for atomic materials, but 15 of which are previously unreported. An illustration of the structural complexity is that one of the previously unreported crystal structures has 100 particles in the unit cell (cI-100-X in Fig. 1), and there are hints of crystals that are even more complex in some yet-to-be-assigned parts of the map. Why is there such structural diversity? For a simple potential, like the Lennard-Jones potential with a single narrow well, the energetically favored structures are those that maximize the number of nearest neighbors with bond distances close to the well bottom. The crystal structures that provide the best solution are the fcc and hexagonal close-packed (hcp) lattices that have coordination environments with 12 equivalent nearest neighbors. However, if an additional repulsive barrier is added to the potential, then a further consideration is minimizing the number of interparticle distances close to this maximum. For example, if the barrier is at roughly $\sqrt{2}$ times the nearest-neighbor distance, fcc and hcp structures become disfavored because of the octahedral interstices in these structures, and, instead, Frank−Kasper crystals, and related dodecagonal quasicrystals, can be stabilized because these structures can be fully divided into tetrahedra (10, 12). Adding a second minimum adds a further constraint for a low-energy structure to satisfy; it must also have sufficient next-nearest distances near to this minimum. Thus, it can be seen that the lowest-energy crystal structure becomes increasingly dependent on the precise set of interparticle distances it possesses, and represents a complex compromise between these different constraints.

Most of the known crystal structures observed in this study correspond to those associated with pure metals or binary intermetallics. For example, as well as the simple fcc, hcp, and body-centered cubic crystals, there are a number of Frank−Kasper phases and structures like $\beta$-tin, $\beta$-manganese, and $\gamma$-brass. This feature is perhaps unsurprising given that the effective pair potential component of metallic interactions can exhibit oscillations (13). By contrast, only a few structures associated with covalent materials were observed, namely, a clathrate and a high-pressure form of silicon (2).

Most of the previously unreported crystal structures have an average coordination number in the intermediate range of 7 to 11 (2). For example, the value for the cI-100-X crystal shown in Fig. 1 is 9.3. There are few known atomic one-component crystals in this coordination number range. Thus, the previously unreported crystals revealed by this study exhibit particular structural novelty.

As well as the diversity of structures, another interesting feature of the crystal structure maps is the relative areas in parameter space occupied by each structure (2). It has been recently shown that many high-dimensional input−output maps (a classical example being a genotype−phenotype map) are highly biased to low-complexity outputs, where the complexity of an output is measured in terms of the minimum information required to describe the output (14).

Although there has been significant interest in interparticle potentials with multiple wells because of their potential for structural complexity, the study of Dshemuchadse et al. provides a systematic survey of the crystal structures that can be formed.

Although the current mapping is only 2D (2), this bias to simplicity is still evident. For example, most of the area of the maps is dominated by relatively simple and well-known crystals, such as bcc, fcc, hcp, A15, $\beta$-tin, cP4, and stacked hexagonal crystals (an exception is the relatively complex Frank−Kasper $\sigma$-phase that has five Wyckoff sites). The previously unreported crystal forms, by contrast, occupy relatively small areas of the parameter space. For a crystal structure, a simple measure of this complexity is the number of Wyckoff sites and the parameters associated with them. By this measure, the previously unreported crystal forms are, on average, about 3 times more complex.

Of course, one hope of such computational studies is that they might reveal potential functional materials with new or improved properties. Therefore, an important question is how realizable might these previously unreported crystal forms be in real systems. Although the competition between interactions with different dependencies on the interparticle separation can lead to multiwell potentials for colloids, it is probably very challenging to fine-tune these potentials to sufficiently match the specific potentials that stabilize these crystal forms, especially given that they occupy such relatively small regions of parameter space. However, the importance of the study is not so much about how precisely to achieve one of these previously unreported crystal forms but rather in its expansion of the space of one-component crystal structures. Indeed, it is hard to imagine how most of these crystals could have been discovered by other means. The study illustrates the power of using large-scale molecular simulations as a means to generate structural novelty (2). Previously unreported crystal forms can then be screened for favorable properties, and those that are promising leads could be used as targets for inverse design approaches (15) with more realizable interactions.

We expect the study of Dshemuchadse et al. (2) will stimulate further exploration of the crystal structures exhibited by multiwell potentials. For example, the study has just focused on the low-pressure behavior, but it would be expected more new crystal forms will be revealed at higher pressures. Similarly, if one considered binary mixtures, the diversity of crystal forms would likely increase significantly.

The study is also likely to be an exemplar for systematic computational explorations of the space of crystal structures associated with other particle properties (Fig. 1). For example, the study of particles with designer directional bonding is likely to be particularly fruitful (16, 17), particularly as DNA origami (18, 19) and de novo designed proteins (20, 21) increasingly offer precise control of their higher-order self-assembly. Such particles could go beyond the geometries allowed by covalent bonding to also reveal many new crystal forms.