Fiveling

A fiveling, also known as a decahedral nanoparticle, a multiply-twinned particle (MTP), a pentagonal nanoparticle, a pentatwin, or a five-fold twin is a type of

Dating back to the nineteenth century there are reports of these particles by authors such as

Most modern analysis started with the observation of these particles by Shozo Ino and Shiro Ogawa in 1966-67,[13]^[14] and independently but slightly later (which they acknowledged) work by John Allpress and John Veysey Sanders.^[15] 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 five single crystal units in the particles, and also the twin relationships. They also observed single crystals and a related type of icosahedral nanoparticle. They called the five-fold and icosahedral crystals multiply twinned particles (MTPs). In the early work near perfect decahedron (pentagonal bipyramid) and icosahedron shapes were formed, so they were called decahedral MTPs or icosahedral MTPs, the names connecting to the decahedral (${\displaystyle D_{5h}}$) and icosahedral (${\displaystyle I_{h}}$) point group symmetries.^[c] Parallel, and apparently independent there was work on larger metal whiskers (nanowires) which sometimes showed a very similar five-fold structure,^[16][17] an occurrence reported in 1877 by Gerhard vom Rath.^[18] There was fairly extensive analysis following this, particularly for the nanoparticles, both of their internal structure by some of the first electron microscopes that could image at the atomic scale,^[19] and by various continuum or atomic models as cited later.

Following this early work there was a large effort, mainly in Japan, to understand what were then called "fine particles", but would now be called nanoparticles. By heating up different elements so atoms evaporated and were then condensed in an inert argon atmosphere, fine particles of almost all the elemental solids were made and then analyzed using electron microscopes. The decahedral particles were found for all face centered cubic materials and a few others, often together with other shapes.^[20][21][22]

While there was some continuing work over the following decades, it was with the National Nanotechnology Initiative^[24] that substantial interest was reignited. At the same time terms such as pentagonal nanoparticle, pentatwin, or five-fold twin became common in the literature, together with the earlier names. A large number of different methods have now been published for fabricating fivelings, sometimes with a high yield but often as part of a larger population of different shapes.^[25] These range from colloidal solution methods^[26] to different deposition approaches.^[20][27] It is documented that fivelings occur frequently for diamond,^[28][29] gold and silver,^[30] sometimes for copper^[31][32] or palladium^[33][34] and less often for some of the other fcc metals such as nickel.^[35] There are also cases such as pentagonite where the crystal structure allows for five-fold twinning with minimal to no elastic strain (see later).^[2] There is work where they have been observed in colloidal crystals consisting of ordered arrays of nanoparticles,^[36][37] and single crystals composed on individual decahedral nanoparticles.^[38] There has been extensive modeling by many different approaches such as embedded atom,^[35] many body,^[39] molecular dynamics,^[40] tight binding approaches,^[41] and density functional theory methods^[42] as recently discussed by Francseca Baletto and Riccardo Ferrando^[43] and also discussed for energy landscapes later.

Disclination strain

These particles consist of five different (single crystal) units which are joined together by twin boundaries. The simplest form shown in the figure has five

Early experimental

The three-dimensional shape depends upon how the fivelings are formed, including the environment such as gas pressure and temperature. In the very early work only pentagonal bipyramids were reported.^[13][14][15] 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. He found a lower energy form where he added {100} facets,^[68] what is now commonly called the Ino decahedron. The surface energy of this form and a related icosahedral twin scale as the two-thirds power of the volume, so they can be lower in energy than a single crystal as discussed further below. However, while Ino was able to explain the icosahedral particles, he was not able to explain the decahedral ones. Later Laurence D. Marks proposed a model using both experimental data and a theoretical analysis, which is based upon a modified Wulff construction which includes more surface facets, including Ino's {100} as well as re-entrant {111} surfaces at the twin boundaries with the possibility of others such as {110}, while retaining the decahedral ${\displaystyle D_{5h}}$ point group symmetry.^[5][6][54] This approach also includes the effect of gas and other environmental factors via how they change the surface energy of different facets. By combining this model with de Wit's elasticity,^[45] Archibald Howie and Marks were able to rationalize the stability of the decahedral to particles.^[54] Other work soon confirmed the shape reported by Marks for annealed particles.^[69] 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,^[35] this name now being widely used.^{[22][30][70][71]}

The minimum energy or thermodynamic shape for these particles^[5][6] 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.^[6][5] An overview of codes to calculate these shapes was published in 2021 by Christina Boukouvala et al.^[72] Considering just {111} and {100} facets:^[5][6]

• The Ino decahedron occurs when the surface energy of the {100} facets is small, ${\displaystyle \gamma _{111}>2\gamma _{100}/{\sqrt {3}}}$;
• Common is the Marks decahedron with {100} facets and a re-entrant surface at the twin boundaries for ${\displaystyle \gamma _{100}/{\sqrt {3}}<\gamma _{111}<2\gamma _{100}/{\sqrt {3}}}$
• With ${\displaystyle \gamma _{111}<\gamma _{100}/{\sqrt {3}}}$ there is no {100} faceting, and the particles have been called nanostars.^[73]
• For very low ${\displaystyle \gamma _{100}}$ the equilibrium shape is a long rod along the common five-fold axis.

The photograph of an 0.5cm gold fiveling from Miass is a Marks decahedron with ${\displaystyle \gamma _{111}\approx 0.85\gamma _{100}}$, while the sketch of Rose^[4] is for ${\displaystyle \gamma _{111}\approx 0.7\gamma _{100}}$. The 75 atom cluster shown above corresponds to the same shape for a small number of atoms. Experimentally, in fcc crystals fivelings with only {111} and {100} facets are common, but many other facets can be present in the Wulff construction leading to more rounded shapes,^[6][70] for instance {113} facets for silicon.^[74] It is known that the surface can reconstruct to a different atomic arrangement in the outermost atomic plane, for instance a dimer reconstruction for {100} facets of silicon particles^[74] of a hexagonal overlayer on the {100} facets of gold decahedra.^[70]

What shape is present depends not just on the surface energy of the different facets, but also upon how the particles grow. The thermodynamic shape is determined by the Wulff construction, which considers the energy of each possible surface facet and yields the lowest energy shape. The original Marks decahedron was based upon a form of Wulff construction that takes into account the twin boundaries.^[5][6] There is a related kinetic Wulff construction where the growth rate of different surfaces is used instead of the energies.^[67][76] This type of growth matters when the formation of a new island on a flat facet limits the growth rate.^[77] If the {100} surfaces of Ino grow faster then they will not appear in the final shape, similarly for the re-entrant surfaces at the twin boundaries -- this leads to the pentagonal bipyramids often observed.^[67] 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.^{[78][79][80][81]}

Another different set of shapes can occur when diffusion of atoms to the particles dominates, a growth regime called diffusion controlled growth. In such cases surface curvature can play a major role,^[83][76] for instance leading to spikes originating at the sharp corners of a pentagonal bipyramids, sometimes leading to pointy stars, as shown in the figure.^[82]

Energy versus size

The most common approach to understand the formation of these particles, first used by Ino in 1969,^[68] is to look at the energy as a function of size comparing icosahedral twins, decahedral nanoparticles and single crystals. The total energy for each type of particle can be written as the sum of three terms:

${\displaystyle E_{total}=E_{surface}V^{2/3}+E_{strain}V+E_{surface\ stress}V^{2/3}}$

for a volume ${\displaystyle V}$, where ${\displaystyle E_{surface}}$ is the surface energy, ${\displaystyle E_{strain}}$ is the disclination strain energy to close the gap (or overlap for marcasite and others), and ${\displaystyle E_{surface\ stress}}$ is a coupling term for the effect of the strain on the surface energy via the surface stress,^[84][85][86] which can be a significant contribution.^[59] 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 an icosahedral particle. Because the decahedral particles have a lower total surface energy than single crystals due (approximately, in fcc) to more low energy {111} surfaces, they are lower in total energy for an intermediate size regime, with the icosahedral particles more stable at very small sizes. (The icosahedral particle have even more {111} surfaces, but also more strain.^[54]) 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.^[53] The very large mineral samples are almost certainly trapped in metastable higher energy configurations.

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.

${\displaystyle Q_{6}}$

${\displaystyle T}$

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[100] who directly observed changes in the internal structure with time during growth. More recent work has observed variations in the internal structure in liquid cells,^[101] or changes between different forms due to either (or both) heating or the electron beam in an electron microscope^{[102][103][104]} including substrate effects.^[40]

Successive Twinning

Allpress and Sanders proposed an alternative approach to energy minimization to understanding these particles called "successive twinning".^[15] Here one starts with a single tetrahedral unit, which then forms a twin either by accident during growth or by collision with another tetrahedron. It was proposed that this could continue to eventually have five units join.

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;^[105] see also Haiqiang Zhao et al^[71] for very similar experimental images. While in many cases experimental images show symmetric structures, sometimes they are less so and the five-fold center is quite asymmetric.^[106][71] There are asymmetric cases which can be metastable,^[5] and asymmetry can also be a strain relief process^[107] or involved in how the particle convert to single crystals or from single crystals.^[100][93] During growth there may be changes, as directly observed by Katsumichi Yagi et al for growth inside an electron microscope,^[100] and migration of the disclination from the outside has been observed in liquid-cell studies in electron microscopes.^[101] 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.^[105]

Connections

There are a number of related concepts and applications of decahedral particles.

Quasicrystals

Soon after the discovery of quasicrystals it was suggested by Linus Pauling^[108][109] that five-fold cyclic twins such as these were the source of the electron diffraction data observed by Dan Shechtman.^[110] While there are similarities, quasicrystals are now considered to be a class of packing which is different from fivelings and the related icosahedral particles.^[111][112]

Heterogeneous catalysts

There are possible links to heterogeneous catalysis, with the decahedral particles displaying different performance.^{[113][114][56][115]} The first study by Avery and Sanders^[113] did not find them in automobile catalysts. Later work by Marks and Howie found them in silver catalysts,^[114] and there have been other reports. It has been suggested that the strain at the surface can change reaction rates,^[56] and since there is evidence that surface strain can change the adsorption of molecules and catalysis there is circumstantial support for this.^[116][117] As of 2024 there is some experimental evidence for different catalytic reactivity.^{[118][115][119]}

Plasmonics

It is known that the response of the surface plasmon polaritons in nanoparticles depends upon their shape.^[120] As a consequence decahedral particles have specific optical responses.^[121][122] One suggested use is to improve light adsorption using their plasmonic properties by adding them to polymer solar cells.^[123]

Thin films and mechanical deformation

Most observations of fivelings have been for isolated particles. Similar structures can occur in thin films when particles merge to form a continuous coating, but do not recrystallize immediately.^[125][126] They can also form during annealing of films,^[127][128] which molecular dynamics simulations have indicated correlates to the motion of twin boundaries and a disclination,^[129] similar to the case of isolated nanoparticles described earlier. There is experimental evidence in thin films for interactions between partial dislocations and disclinations,^[130] as discussed in 1971 by de Wit.^[44] They can also be formed by mechanical deformation.^[124] The formation of a local fiveling structure by annealing or deformation has been attributed to a combination of stress relief and twin motion,^{[127][124][131]} which is different from the surface energy driven formation of isolated particles described above.

See also

• Cluster (chemistry)
   – Collection of bound atoms or moleculesPages displaying short descriptions of redirect targets
• Cluster (physics) – Small collection of atoms or molecules
• Crystal habit – Mineralogical term for the visible shape of a mineral
• Crystal twinning – Two separate crystals sharing some of the same crystal lattice points in a symmetrical manner
• Disclination – Angular defect in a material
• Icosahedral twins – Structure found in atomic clusters and nanoparticles
• Nanocluster – Collection of bound atoms or molecules
• Nanomaterials – Materials whose granular size lies between 1 and 100 nm
• Nanowire – Wire with a diameter in the nanometres
• Nucleation – Initial step in the phase transition or molecular self-assembly of a substance
• Surface energy – Excess energy at the surface of a material relative to its interior
• Surface stress – Change of surface energy with strain
• Wulff construction – Lowest energy shape of a single crystal

External links

• "Crystal creator code". www.on.msm.cam.ac.uk. Retrieved 2024-04-01. Code from the group of Emilie Ringe which calculates thermodynamic and kinetic shapes for decahedral particles and also does optical simulations, see also.^[72]
• "WulffPack – a package for Wulff constructions". wulffpack.materialsmodeling.org. Retrieved 2024-04-01. Code from J M Rahm and P Erhart which calculates thermodynamic shapes, both continuum and atomistic, see also.^[132]

Notes

References