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. 2007 Nov 20;104(47):18382-6.
doi: 10.1073/pnas.0703431104. Epub 2007 Nov 14.

Faceting Ionic Shells Into Icosahedra via Electrostatics

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Free PMC article

Faceting Ionic Shells Into Icosahedra via Electrostatics

Graziano Vernizzi et al. Proc Natl Acad Sci U S A. .
Free PMC article

Abstract

Shells of various viruses and other closed packed structures with spherical topology exhibit icosahedral symmetry because the surface of a sphere cannot be tiled without defects, and icosahedral symmetry yields the most symmetric configuration with the minimum number of defects. Icosahedral symmetry is different from icosahedral-shaped structures, which include some large viruses, cationic-anionic vesicles, and fullerenes. We present a faceting mechanism of ionic shells into icosahedral shapes that breaks icosahedral symmetry resulting from different arrangements of the charged components among the facets. These self-organized ionic structures may favor the formation of flat domains on curved surfaces. We show that icosahedral shapes without rotational symmetry can have lower energy than spheres with icosahedral symmetry caused by preferred bending directions in the planar ionic lattice. The ability to create icosahedral shapes without icosahedral symmetry may lead to the design of new functional materials. The electrostatically driven faceting mechanism we present here suggests that we can design faceted polyhedra with diverse symmetries by coassembling oppositely charged molecules of different stoichiometric ratios.

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
A snapshot of an unrelaxed distribution of positive and negative charges over a sphere and an icosahedron (Left). The triangular lattice on the Right shows some examples of the quasi-equivalent triangulations of the icosahedron; the colored triangles represent a face of the icosahedra with T = 3, 13, and 31 (with total number of particles n = 32, 132, and 312, respectively).
Fig. 2.
Fig. 2.
Examples of charge distributions with spherical topology and icosahedral shape. From right to left, top to bottom, the number of particles N is 32, 132, 212, and 312 (T is 3, 13, 21, and 31), respectively, for 3:1 stoichiometric ratio (+3 charges are red, −1 charges are blue). Note that in the n = 32 (T = 3) case, the eight “+3” charges are on the vertices of a regular cube and not on a square antiprism as in the Thomson problem for eight charges on a sphere.
Fig. 3.
Fig. 3.
The projection of an icosahedron onto the sphere is not unique. (Left) All of the 20 faces of the icosahedron are projected onto spherical triangles. (Right) Every point P of the triangular face ABC is mapped onto a point P′ on the sphere (a) (in red). The gnomic projection (b) is not as uniform as our equal-area projection (c).
Fig. 4.
Fig. 4.
The relative electrostatic energy difference between icosahedral configurations and spherical configurations versus the total number of charges (3:1 and 5:1 cases, in red and blue, respectively); the points below the solid line correspond to systems with icosahedral shapes.

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