Defective ground states of toroidal crystals

Crystalline assemblages of identical subunits packed together and elastically bent in the form of a torus have been found in the past ten years in a variety of systems of surprisingly different nature, such as viral capsids, self-assembled monolayers, and carbon nanomaterials. In this Rapid Communication we analyze the structural properties of toroidal crystals and provide a unified description based on the elastic theory of defects in curved geometries. We find ground states characterized by the presence of fivefold disclinations on the exterior of the torus and sevenfold disclinations in the interior. The number of excess disclinations is controlled primarily by the aspect ratio of the torus, suggesting a mechanism for creating toroidal templates with precisely controlled valency via functionalization of the defect sites.

Figure 1

(Color online) Isolated defects and scar phases in the $(r,V)$ plane. When the number of vertices $V$ increases, the range of the screening curvature becomes smaller than one lattice spacing and disclinations appear, delocalized in the form of a 5-7-5 grain boundary miniscar.

Figure 2

(Color online) Elastic energy of a 5-7 disclination dipole constrained to lie on the same meridian, as a function of the angular separation. In the inset, illustration of a circular torus of radii $R_1>R_2$. Regions of positive and negative Gaussian curvature have been shaded in red and blue, respectively. A standard parametrization of the torus is obtained by considering the angles $\psi$ and $\varphi$.

Figure 3

(Color online) Phase diagram for $T_p$ configurations in the plane $(r,{ϵ}_c∕AY)$. For $r∊[3.68,10.12]$ and ${ϵ}_c\sim 0$ the structure is given by a $T_{10}$ configuration with symmetry group $D_{5h}$.

Figure 4

(Color online) Phase diagram of a fivefold-symmetric lattice in the plane $(r,\delta \psi )$. For small $\delta \psi$ and $r$ in the range [3.3,7.5], the prismatic TP5 configuration is energetically favored. For $r<3.3$, the system undergoes a structural transition to the antiprismatic phase TA5.