Shape Selection in Chiral Self-Assembly

Many biological and synthetic materials self-assemble into helical or twisted aggregates. The shape is determined by a complex interplay between elastic forces and the orientation and chirality of the constituent molecules. We study this interplay through Monte Carlo simulations, with an accelerated algorithm motivated by the growth of an aggregate out of solution. The simulations show that the curvature changes smoothly from cylindrical to saddlelike as the elastic moduli are varied. Remarkably, aggregates of either handedness form from molecules of a single handedness, depending on the molecular orientation.

Figure 1

Model of a chiral elastic ribbon as a tethered membrane composed of interaction sites connected by elastic springs to form a triangular lattice. The local molecular orientation at each site is described by a director, as indicated by the arrows.

Figure 2

Three views of a helical ribbon, showing the 3D structure. The ribbon has grown to a size of $16\times 216$ sites.

Figure 3

(a) Series of ribbons grown with the parameter $K_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{h}}=2$, 4, 6, 8, and 10, showing the crossover from twisted ribbons with Gaussian saddlelike curvature to helical ribbons with cylindrical curvature. (b) Plot of the Gaussian curvature for the five ribbons, calculated at the central point of the ribbons, in units of the hard-core diameter $a^{-2}$. This plot confirms that all of the structures have some Gaussian curvature, and that the level of Gaussian curvature changes gradually as a function of the elastic parameters.

Figure 4

Two ribbons grown with the same elastic parameters, but with different orientations of the director representing the molecular tilt. In the left case, the tilt is aligned along the long axis of the ribbon, but in the right case it is aligned along the short axis. This change in the tilt direction reverses the handedness of the ribbon, and greatly changes the radius and pitch.