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

TL;DR

This paper investigates how truncated tetrahedra self-assemble into various crystal structures driven by directional entropic forces, revealing new insights into the role of shape in entropy-driven assembly.

Contribution

It introduces a comprehensive study of thermodynamic self-assembly of truncated tetrahedra, identifying new crystal structures and elucidating the influence of shape on entropic forces.

Findings

Identification of several stable crystal structures including diamond and β-tin.

Discovery of a new space-filling polyhedron overlooked in previous research.

Demonstration that self-assembled structures are governed by face-to-face alignment tendencies.

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, {\beta}-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.