Controlling Chirality of Entropic Crystals

Colloidal crystal structures with complexity and diversity rivaling atomic and molecular crystals have been predicted and obtained for hard particles by entropy maximization. However, thus far homochiral colloidal crystals, which are candidates for photonic metamaterials, are absent. Using Monte Carlo simulations we show that chiral polyhedra exhibiting weak directional entropic forces self-assemble either an achiral crystal or a chiral crystal with limited control over the crystal handedness. Building blocks with stronger faceting exhibit higher selectivity and assemble a chiral crystal with handedness uniquely determined by the particle chirality. Tuning the strength of directional entropic forces by means of particle rounding or the use of depletants allows for reconfiguration between achiral and homochiral crystals. We rationalize our findings by quantifying the chirality strength of each particle, both from particle geometry and potential of mean force and torque diagrams.

Figure 1

Assembly of the $\beta$-Mn crystal structure from its VPs. (a) The unit cell with 20 particles. The particles form two intertwined chiral square helices in the direction of the fourfold axis. (b) The VP shown on the left (orange) is elongated with 14 faces and 22 vertices. The VP shown on the right (red) resembles a dodecahedron with 12 irregular pentagonal faces and 20 vertices. (c) We observe the assembly of both left-handed and right-handed variants, with a bias for right-handed, the handedness dictated by the VPs. (d) A snapshot of the assembled crystal with a unit cell highlighted. The bond-order diagram (inset) allows identifying the crystal structure. The crystal is degenerate, which means either of the two VPs can be found at any lattice point with comparable probability. (e) A helix cut out from the assembled crystal. The VPs (shown at 75% of their original size) in the double-square helix [see (a)] are orientationally disordered. Note that the double helix can also be interpreted as a tetrahelix of opposite handedness, as indicated in the figure.

Figure 2

Assembly of the simple chiral cubic (scc) crystal structure from its VPs. (a) A $2\times 2\times 2$ patch of the unit cell with 4 particles. The bonds indicate chiral square helices (pitch length four). (b) The VP has point group $D_3$ and six symmetry-equivalent big faces. (c) The assembled crystal is homochiral with handedness determined by the building block. (d) A snapshot of the assembled crystal structure. (e) A right-handed helix cut out from the assembled crystal. The VPs align their big faces in the square helix [see (a)]. As the crystal is cubic, similar square helices are found along the other two fourfold axes.

Figure 3

Reconfiguration between an achiral crystal and a homochiral crystal. We compare the assembly of VPs of the scc crystal (a) using their original polyhedral shape, (b) with rounded edges and corners, and (c) additionally in the presence of depletants. (d) Rounding weakens DEFs and leads to the assembly of a bcc rotator crystal. (e) The effect of rounding can be reversed by depletion, which recovers scc. Depletants were simulated implicitly (see text for details). They are sketched in (c) and omitted in (e).

Figure 4

Quantifying the chiral strength of the particles used in this study. (a)–(d) PMFT diagrams for the four systems. The PMFT diagrams were calculated by binning the frequency of finding two particle centers at a given separation vector, creating a histogram whose negative logarithmic value corresponds to the energy units reported [32, 37]. The diagrams are two-dimensional slices taken through the first neighbor shell parallel to the largest facet on a shape. They were obtained at the lowest densities at which each crystal was observed to self-assemble. Color indicates the PMFT, from $0kT$ (dark blue) to $-6kT$ (dark red). (e) Coordination number versus isoperimetric quotient plot derived from Ref. [30] including the shapes from (a)–(d).