Pressure-tunable photonic band gaps in an entropic colloidal crystal

Materials adopting the diamond structure possess useful properties in atomic and colloidal systems and are a popular target for synthesis in colloids where a photonic band gap is possible. The desirable photonic properties of the diamond structure pose an interesting opportunity for reconfigurable matter: Can we create a crystal able to switch reversibly to and from the diamond structure with a photonic band gap in the visible light range? Drawing inspiration from high-pressure transitions of diamond-forming atomic systems, we design a system of polyhedrally shaped colloidal particles with spherical cores that transitions from diamond to a tetragonal diamond derivative upon a small pressure change. The transition can alternatively be triggered by changing the shape of the particle in situ. We propose that the transition provides a reversible reconfiguration process for a potential new colloidal material and draw parallels between this transition and the phase behavior of the atomic transitions from which we take inspiration.

Figure 1

The ${\mathrm{\Delta }}_{323}$ shape parameterization. (a) and (b) The ${\mathrm{\Delta }}_{323}$ shape family, ranging from an octahedron $\left(\alpha =0.0\right)$ to a tetrahedron $\left(\alpha =1.0\right)$. The intersection of planes at parametrized distances shown in (a) generate the resulting shapes in (b). All shapes are scaled to unit volume in simulations.

Figure 2

Structural diagrams for diamond and its tetragonal derivative. (a)–(d) Structural comparison between (a) and (c) diamond and (b) and (d) the TDD with $c/a=\sqrt{0.4}$, both shown with the shapes at $\alpha =0.5$. Here, we show projections along the major axis of cubic diamond and along two differing axes in the tetragonal derivative structure $[\vec{c}$ in (b) and one of the equivalent $\vec{a}$ axes in (d)]. The particle positions appear unchanged when viewed along the $\vec{c}$ axis, but the particles rotate about the $\vec{c}$ direction. (e) The peaks in the radial distribution function (RDF), i.e., the distances of the nearest-neighbor shells. The fourth-nearest neighbors (yellow) in the $\vec{c}$ direction in diamond are immediately outside the third-neighbor shell in TDD at $c/a=\sqrt{0.4}$.

Figure 3

Simulation results for $NVT\alpha ,NPT\alpha$, and $NVT\mu$. (a) Composite phase diagram from all simulations. The putative densest packing in either the diamond or the TDD structure as a function of shape is the upper limit of the phase diagram, the geometrically forbidden region shown in gray. Shades of red represent the $c/a$ ratio of the unit cell with the lighter red region showing where the diamond is the configuration with the lowest free energy $\left(c/a=1\right)$ and darker reds indicating TDD structures with lower $c/a$ values. The black data set represents results from $NVT\mu$ simulations, denoting where the crossover from the diamond to the TDD stability region occurs as a function of density and shape. (b) Equation of state for $\alpha =0.5$ evaluated in compression and expansion runs. The main plot shows the average values with an inset showing the average and standard deviations around the transition pressure. (c) Potentials of mean force and torque for $\alpha =0.5$ at varying pressures. The PMFT wells shift from the center of the large face of each particle toward one edge. This shows that the coordination remains tetrahedral but becomes distorted with increasing pressure.

Figure 4

Photonic band-gap structures and sizes. (a) Size of photonic band gaps between bands 2 and 3 (red) and bands 8 and 9 (blue) for structures in Fig. 3. All polyhedra have been replaced with spheres with radii equal to the polyhedral insphere radii and with dielectric constant $\epsilon =11.56$. (b) and (c) Representative photonic band structures for $\alpha =0.5$ at (b) $\varphi =0.6$ where it is in cubic diamond and (c) $\varphi =0.95$ where it is in TDD at $\approx \sqrt{0.4}$. The insphere radius of ${\mathrm{\Delta }}_{323}{|}_{\alpha =0.5}$ is 0.425.