Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces

Abstract

Polyhedra and their arrangements have intrigued humankind since the ancient Greeks and are today important motifs in condensed matter, with application to many classes of liquids and solids. Yet, little is known about the thermodynamically stable phases of polyhedrally shaped building blocks, such as faceted nanoparticles and colloids. Although hard particles are known to organize due to entropy alone, and some unusual phases are reported in the literature, the role of entropic forces in connection with polyhedral shape is not well understood. Here, we study thermodynamic self-assembly of a family of truncated tetrahedra and report several atomic crystal isostructures, including diamond, β-tin, and high-pressure lithium, as the polyhedron shape varies from tetrahedral to octahedral. We compare our findings with the densest packings of the truncated tetrahedron family obtained by numerical compression and report a new space-filling polyhedron, which has been overlooked in previous searches. Interestingly, the self-assembled structures differ from the densest packings. We show that the self-assembled crystal structures can be understood as a tendency for polyhedra to maximize face-to-face alignment, which can be generalized as directional entropic forces.

Crystal-structure prediction via the Floppy-Box Monte Carlo algorithm: Method and application to hard (non)convex particles

“…As previously mentioned, similar approaches have been applied for tetrahedra 65,90 and truncated tetrahedra. 68,91 However, it is also possible that the numerical results are of sufficient quality to "guess" the analytic solutions which maximizes the density and verify that this structure does not give overlaps using an analytic form of our overlap techniques. We used this approach for rhombicuboctahedra and rhombic enneacontrahedra in Ref.…”

“…For the truncated tetrahedra, we discovered a new crystal structure which improved upon the literature value of the densest packing. Subsequent numerical and analytic studies 68,91 showed that this value could be further improved upon.…”

“…28 This has led to particular interest from the materials science community in these overlap algorithms to perform simulations on nanoparticle and colloid systems. [63][64][65][66][67][68][69][70][71] A. The method of separating axes…”

“…43 However, establishing the structure of a numerically obtained result is highly nontrivial in general. Moreover, a description of a system by the atomic equivalent of the lattice on which the particles' centres of rotation reside, 43,68,69,88 is not always adequate for nonspherical particles, because of the strong relevance of the orientation of the particles. 70 Finally, we note that the frequency with which structures are observed in the FBMC runs can give some insight into which of the structures we find is stable, 43 but only free-energy calculations can give a definitive answer.…”

“…Currently, several other techniques exist to estimate this lower bound numerically. 45,63,65,68,71,89,92,93 The FBMC method is similar to the adaptive-shrinking-cell (ASC) method of Refs. 63 and 89, since both allow for a sequential search of configurational space and lattice space using a Metropolis based MC procedure.…”