Density functional theory for hard polyhedra

The ready availability of polyhedral nanoparticles and colloids as a result of recent advances in synthesis methods have allowed experimental observations of crystals of polyhedra. While this realization of polyhedra spurred simulation work on the structure of hard particles, theoretical approaches, so far, have been limited. Using the framework of geometry-based fundamental-measure theory, we develop a classical density functional (DFT) for hard polyhedra and their mixtures. We apply the DFT to Platonic solids (tetrahedra, cubes, octahedra, dodecahedra and icosahedra) and perform Monte Carlo simulations.

Knowledge of the structure of colloids or nanoparticles near a wall is important for understanding heterogeneous nucleation. Furthermore, the density profile near a hard wall provides a standard test case for DFT by comparing with simulation results. The faceted shape of the polyhedra leads to complex layering and orientational ordering near the wall already for a one-component system which is excellently reproduced by our theory. We also considered a 2:1 mixture of tetrahedra and octahedra, which is interesting because it can form a close packed crystal with unit packing fraction. Surprisingly, the local structure in the fluid near the wall for this binary mixture is not similar to the best-packed structure. These effects can be verified in real-space experiments on polyhedral colloids.