Density Functional Theory for Hard Polyhedra

Using the framework of geometry-based fundamental-measure theory, we develop a classical density functional for hard polyhedra and their mixtures and apply it to inhomogeneous fluids of Platonic solids near a hard wall. As revealed by Monte Carlo simulations, the faceted shape of the polyhedra leads to complex layering and orientational ordering near the wall, which is excellently reproduced by our theory. These effects can be verified in real-space experiments on polyhedral colloids.

Figure 1

The polyhedra considered in this work are tetrahedra, cubes (hexahedra), octahedra, dodecahedra, and icosahedra with (equal) edge lengths $l$ as indicated. The different colors are simply to allow facile distinction of the faces.

Figure 2

Center-of-mass density profiles as a function of the height $z$ divided by the edge length $l$ for one-component systems of (a) tetrahedra, (b) cubes, (c) octahedra, (d) dodecahedra, and (e) icosahedra, which are depicted in Fig. 1. Fundamental-measure DFT (dark or blue lines) is compared to results from Monte Carlo (MC) simulations (light or orange lines). The average volume fraction $\eta \equiv v_{p}N/AH$ is set to $\eta =0.3$, where $v_p$ is the volume of the particles. (f) The density profile of spheres for the same packing fraction $\eta =0.3$ is shown as a function of the height $z$ divided by the diameter $\sigma$ for comparison. The heights for which a particle can just touch the wall with a vertex, an edge, and a face are indicated by the dashed lines; also shown are polyhedra with the corresponding orientations.

Figure 3

(a)–(e) Orientational order parameter profiles $P_1(z)$, which measure the degree of face-wall alignment in one-component systems for the Platonic solids (see Fig. 1) as a function of the dimensionless distance to the wall $z/l$. DFT (dark or blue lines) is compared to MC simulation results (light or orange lines). The gray rectangles denote the $z$ ranges that are inaccessible due to overlap with the wall for all orientations. The arrows denote the $z$ positions of the peaks in the center-of-mass density profile (Fig. 2). (f) The dimensionless wall-fluid interfacial tensions $\gamma {\sigma }_{\mathrm{eff}}^2/k_{B}T$ from DFT calculations for the Platonic polyhedra with $n_F$ faces and spheres ($n_F=\infty$) [27], where ${\sigma }_{\mathrm{eff}}$ is the diameter of a sphere with the same volume as the particle of interest.

Figure 4

Partial center-of-mass density profiles ${\eta }_s(z)$ for a $2:1$ mixture of tetrahedra (dark or blue lines) for $s=1$ and octahedra (light or orange lines) for $s=2$ with equal edge lengths $l$ as a function of the dimensionless distance to the wall $z/l$. The symbols, lines, and arrows of the same color denote the simulation results for the mixture, the theory for the mixture, and the $z$ positions of the bulk $\eta (z)$ peaks, respectively, of the corresponding species. The results should be compared to those in the bulk; see Fig. 2a for tetrahedra and Fig. 2c for octahedra. The inset shows an example configuration as expected from expanding a close-packed layer.