Fiveling

Information about them is distributed across a diverse range of scientific disciplines, mainly chemistry, materials science, mineralogy, nanomaterials and physics.

Dating back to the nineteenth century there are reports of these particles by authors such as Jacques-Louis Bournon in 1813 for marcasite,[9][10] and Gustav Rose in 1831 for gold.

[13] The older literature was mainly observational, with information on many materials documented by Victor Mordechai Goldschmidt in his Atlas der Kristallformen.

Using transmission electron microscopy and diffraction these authors demonstrated the presence of the five single crystal units in the particles, and also the twin relationships.

[c] Parallel, and apparently independent there was work on larger metal whiskers (nanowires) which sometimes showed a very similar five-fold structure,[19][20] an occurrence reported in 1877 by Gerhard vom Rath.

At the same time terms such as pentagonal nanoparticle, pentatwin, or five-fold twin became common in the literature, together with the earlier names.

[23][30] It is documented that fivelings occur frequently for diamond,[31][32] gold and silver,[33] sometimes for copper[34][35] or palladium[36][37] and less often for some of the other face-centered cubic (fcc) metals such as nickel.

At small sizes this gap is closed by an elastic deformation, which Roland de Wit pointed out[46][47] could be described as a wedge disclination, a type of defect first discussed by Vito Volterra in 1907.

[56] More recently there has been detailed analysis of the atomic positions first by Craig Johnson et al,[57] followed up by a number of other authors,[58][59][60] providing more information on the strains and showing how they are distributed in the particles.

[62] In addition, as pointed out by Srikanth Patala, Monica Olvera de la Cruz and Marks[51] and shown in the figure, the Von Mises stress are different for (kinetic growth) pentagonal bipyramids versus the minimum energy shape.

[68] However, these x-ray measurements only see the average which necessarily shows a tetragonal arrangement, and there is extensive evidence for inhomogeneous deformations dating back to the early work of Allpress and Sanders,[18] Tsutomu Komoda,[22] Marks and David J. Smith[52] and more recently by high resolution imaging of details of the atomic structure.

[16][17][18] In 1970 Ino tried to model the energetics, but found that these bipyramids were higher in energy than single crystals with a Wulff construction shape.

By combining this model with de Wit's elasticity,[47] Archibald Howie and Marks were able to rationalize the stability of the decahedral to particles.

[71] This was further confirmed in detailed atomistic calculations a few years later by Charles Cleveland and Uzi Landman who coined the term Marks decahedra for these shapes,[4] this name now being widely used.

[25][33][72][73] The minimum energy or thermodynamic shape for these particles[7][8] depends upon the relative surface energies of different facets, similar to a single crystal Wulff shape; they are formed by combining segments of a conventional Wulff construction with two additional internal facets to represent the twin boundaries.

[8][7] An overview of codes to calculate these shapes was published in 2021 by Christina Boukouvala et al.[74] Considering just {111} and {100} facets:[7][8] The photograph of an 0.5 cm gold fiveling from Miass is a Marks decahedron with

The original Marks decahedron was based upon a form of Wulff construction that takes into account the twin boundaries.

[69] Alternatively, if the {111} surfaces grow fast and {100} slow the kinetic shape will be a long rod along the common five-fold axis as shown in the figure.

[83] The most common approach to understand the formation of these particles, first used by Ino in 1969,[70] is to look at the energy as a function of size comparing icosahedral twins, decahedral nanoparticles and single crystals.

[88][89][90] In addition, as first described by Michael Hoare and P Pal[91] and R. Stephen Berry[92][93] and analyzed for these particles by Pulickel Ajayan and Marks[94] as well as discussed by others such as Amanda Barnard,[95] David J. Wales,[41][64][96] Kristen Fichthorn[97] and Baletto and Ferrando,[45] at very small sizes there will be a statistical population of different structures so many different ones will coexist.

In many cases nanoparticles are believed to grow from a very small seed without changing shape, and reflect the distribution of coexisting structures.

[26] There has been experiment support based upon work where single nanoparticles are imaged using electron microscopes either as they grow or as a function of time.

One of the earliest works was that of Yagi et al[101] who directly observed changes in the internal structure with time during growth.

[42] Allpress and Sanders proposed an alternative approach to energy minimization to understanding these particles called "successive twinning".

[106] The term "successive twinning" has now come to mean a related concept: motion of the disclination either to or from a symmetric position as sketched in the atomistic simulation in the figure;[106] see also Haiqiang Zhao et al[73] for very similar experimental images.

[102] Extensive details about the atomic processes involved in motion of the disclination have been given using molecular dynamics calculations supported by density functional theory as shown in the figure.

Soon after the discovery of quasicrystals it was suggested by Linus Pauling[109][110] that five-fold cyclic twins such as these were the source of the electron diffraction data observed by Dan Shechtman.

Similar structures can occur in thin films when particles merge to form a continuous coating, but do not recrystallize immediately.

[126][127] They can also form during annealing of films,[128][129] which molecular dynamics simulations have indicated correlates to the motion of twin boundaries and a disclination,[130] similar to the case of isolated nanoparticles described earlier.

There is experimental evidence in thin films for interactions between partial dislocations and disclinations,[131] as discussed in 1971 by de Wit.