Icosahedral twins

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FCC icosahedral model projected down the 5-fold on the left and 3-fold zone axis orientation on the right. IcotwinModel.png
FCC icosahedral model projected down the 5-fold on the left and 3-fold zone axis orientation on the right.

An icosahedral twin is a nanostructure found in atomic clusters and also nanoparticles with some thousands of atoms. 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, [1] [2] [3] 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, chemical synthesis) lead to the icosahedral form, and they also occur in virus capsids.

Contents

They occur at small sizes where they have lower surface energies than other configurations. This is balanced by a 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.

Energetics

In a large particle the form that it takes is dominated by the bulk bonding, leading to a Wulff construction shape. However, when the size is reduced there are a significant number of atoms at the surface, and hence the surface energy starts to become important. Icosahedral arrangements, typically because of their smaller surface energy, [4] may be preferred for small clusters. For face centered cubic 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. The external surface shape can be generated from a modified Wulff construction. [3] and is also not always that of a simple icosahedron; there can be additional facets at the twin boundaries leading to a more spherical shape. [3] There are several software codes that can be used to calculate the shape as a function if the energy of different surface facets. [5] [6]

With just tetrahedra these structure cannot fill space and there would be gaps, so there is some distortions of the atomic positions, that is elastic deformation to close these gaps. [4] Roland De Wit pointed out that these can be thought of in terms of disclinations, [7] an approach later extended to three dimensions by Elisabeth Yoffe. [8] This leads to a compression in the center of the particles, and an expansion at the surface. [8]

At larger sizes the energy to distort becomes larger than the gain in surface energy, and bulk materials (i.e. sufficiently large clusters) generally revert to one of the crystalline close-packing configurations. In principle they will convert to a simple single crystal with a Wulff construction [9] shape. The size when they become less energetically stable is typically in the range of 10-30 nanometers in diameter, [10] but it does not always happen that the shape changes and the particles can grow to micron sizes. [11]

The most common approach to understand the formation of these particles, first used by Shozo Ino in 1969, [4] 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:

Energy landscape for a 75 atom Leonard-Jones cluster for temperature and an order parameter. Energy landscape for 75 atom Leonard-Jones cluster.tif
Energy landscape for a 75 atom Leonard-Jones cluster for temperature and an order parameter.

for a volume , where is the surface energy, is the disclination strain energy to close the gap , and is a coupling term for the effect of the strain on the surface energy via the surface stress, [13] [14] [15] which can be a significant contribution. [16] The sum of these three terms is compared to the total surface energy of a single crystal (which has no strain), and to similar terms for a decahedral particle. Of the three the icosahedral particles have both the lowest total surface energy and the largest strain energy for a given volume. Hence the icosahedral particles are more stable at very small sizes. At large sizes the strain energy can become very large, so it is energetically favorable to have dislocations and/or a grain boundary instead of a distributed strain. [17]

There is no general consensus on the exact sizes when there is a transition in which type of particle is lowest in energy, as these vary with material and also the environment such as gas and temperature; the coupling surface stress term and also the surface energies of the facets are very sensitive to these. [18] [19] [20] In addition, as first described by Michael Hoare and P Pal [21] and R. Stephen Berry [22] [23] and analyzed for these particles by Pulickel Ajayan and Laurence Marks [24] as well as discussed by others such as Amanda Barnard, [25] David J. Wales, [26] [27] [28] Kristen Fichthorn [29] and Francesca Baletto and Riccardo Ferrando, [30] at very small sizes there will be a statistical population of different structures so many different ones will exist at the same time. In many cases nanoparticles are believed to grow from a very small seed without changing shape, and hence what is found reflects the distribution of coexisting structures. [3]

For systems where icosahedral and decahedral morphologies are both relatively low in energy, the competition between these structures has implications for structure prediction and for the global thermodynamic and kinetic properties. These result from a double funnel energy landscape [31] [32] where the two families of structures are separated by a relatively high energy barrier at the temperature where they are in thermodynamic equilibrium. This arises for a cluster of 75 atoms with the Lennard-Jones potential, where the global potential energy minimum is decahedral, and structures based upon incomplete Mackay icosahedra [33] are also low in potential energy, but higher in entropy. The free energy barrier between these families is large compared to the available thermal energy at the temperature where they are in equilibrium. An example is shown in the figure, with probability in the lower part and energy above with axes of an order parameter and temperature . At low temperature the 75 atom decahedral cluster (Dh) is the global free energy minimum, but as the temperature increases the higher entropy of the competing structures based on incomplete icosahedra (Ic) causes the finite system analogue of a first-order phase transition; at even higher temperatures a liquid-like state is favored. [12]

Ubiquity

Electron micrograph of two Icosahedral adenoviruses, with an illustration to show the shape. Icosahedral Adenoviruses.jpg
Electron micrograph of two Icosahedral adenoviruses, with an illustration to show the shape.

Most modern analysis of these shapes in nanoparticles started with the observation of icosahedral and decahedral particles by Shozo Ino and Shiro Ogawa in 1966-67, and independently but slightly later (which they acknowledged) in work by John Allpress and John Veysey Sanders. In both cases these were for vacuum deposition of metal onto substrates in very clean (ultra-high vacuum) conditions, where nanoparticle islands of size 10-50 nm were formed during thin film growth. Using transmission electron microscopy and diffraction these authors demonstrated the presence of the units in the particles, and also the twin relationships. They called the five-fold and icosahedral crystals multiply twinned particles (MTPs). In the early work near perfect icosahedron shapes were formed, so they were called icosahedral MTPs, the names connecting to the icosahedral () point group symmetry.These forms occur for both elemental nanoparticles [34] [35] as well as alloys [36] [37] and colloidal crystals. [38] A related form also exists in icosahedral viruses. [39] [40]

Quasicrystals are un-twinned structures with long range rotational but not translational periodicity, that some initially tried to explain away as icosahedral twinning. [41]

See also

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<span class="mw-page-title-main">Fiveling</span> Five crystals arranged round a common axis

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References

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