Icosahedral twins

An icosahedral twin is an atomic structure found in atomic clusters and also nanoparticles with some thousands of atoms. Their atomic structure is slightly different from what is found for bulk materials, and contains five-fold symmetries. They have been analyzed in many areas of science including crystal growth, crystallography, chemical physics, surface science and materials science, and are sometimes considered as beautiful due to their high symmetry. The simplest form of these clusters is twenty interlinked tetrahedral crystals joined along triangular (e.g. cubic-(111)) faces, although more complex variants of the outer surface also occur. A related structure has five units similarly arranged with twinning, which were known as "fivelings" in the 19th century, and more recently as "decahedral multiply twinned particles", "pentagonal particles" or "star particles". A variety of different methods (e.g. condensing metal nanoparticles in argon, deposition on a substrate, wet chemical synthesis) lead to the icosahedral form, and they also occur in virus capsids. These forms occur at small sizes where they have lower total surface energy than other configurations. This is balanced by an elastic deformation (strain) energy, which dominates at larger sizes. This leads to a competition between different forms as a function of size, and often there is a population of different shapes.

Shape and energetics In a large particle the energy is dominated by the bulk bonding. The energy of the external surface where the atoms have less bonding is less important. The overall shape is the one which minimizes the total surface energy, the solution of which is the Wulff construction. When the size is reduced a significant fraction of the atoms are at the surface, and hence the total surface energy starts to become comparable to the bulk bonding energy. Icosahedral arrangements, typically because of their smaller total surface energy, can be preferred for small nanoparticles. For face centered cubic (fcc) materials such as gold or silver these structures can be considered as being built from twenty different single crystal units all with three twin facets arranged in icosahedral symmetry, and mainly the low energy {111} external facets. An fcc single crystal has both {111} and {100} surface facets, and perhaps {110} if the energy of the latter is low enough. In contrast icosahedral twins normally have {111} and perhaps {110}, none of the higher energy {100}.

The external surface shape for given values of the surface energy can be generated from a modified Wulff construction, and is not always that of a simple icosahedron; there can be additional facets leading to a more spherical shape as illustrated in the figure. Depending upon the relative energies of {111} and {110} facets, the shape can range from an icosahedron (on the left of the figure) with small dents at the five-fold axes (due to the twin boundary energy) when {111} is significantly lower in energy, to (going to the right in the figure) a truncated icosahedron or an icosidodecahedron when the {111} and {110} are similar, and a regular dodecahedron when {110} is significantly lower in energy. The limit where the arrangement of atoms leads to a regular icosahedron is often called a MacKay icosahedron; there can also be a reconstruction of some of the surface atoms to a hexagonal coordination, which is called an anti-MacKay icosahedron. These different shapes have been found in experiments where the relative surface energies are changed with surface adsorbates. There are several software codes that can be used to calculate the shape as a function of the energy of different surface facets. Packing rules for various types of icosahedra with multiple components are also known.

Made out of single crystal fcc units, these structure cannot fill space and there would be gaps as shown in the figure, so there are some distortions of the atomic positions, equivalent to an elastic deformation to close these gaps. These deformations cost energy, and this strain energy competes with the gain in total surface energy. Roland De Wit pointed out that these can be thought of in terms of disclinations, an approach later extended to three dimensions by Elisabeth Yoffe. This leads to a compression in the center of the particles, and an expansion at the surface. At small sizes the surface energy often dominates over the strain energy, with icosahedral forms often the most stable ones. At larger sizes the energy to distort becomes larger than the gain in surface energy, and a single crystal with a Wulff construction shape is lowest in energy. The size when the icosahedra become less energetically stable is typically 10-30 nanometers in diameter, but it does not always happen that the shape changes and the particles can grow to micron sizes. The most common approach to understand the formation of these particles, first used by Shozo Ino in 1969, is to look at the energy as a function of size comparing these icosahedral twins, decahedral nanoparticles and single crystals. The total energy for each type of particle can be written as the sum of three terms:

E t o t a l = E s u r f a c e V 2 / 3 + E s t r a i n V + E s u r f a c e s t r e s s V 2 / 3 {\displaystyle E_{total}=E_{surface}V^{2/3}+E_{strain}V+E_{surface\ stress}V^{2/3}}

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