Extended Wulff constructions

Extended Wulff constructions refers to a number of different ways to model the structure of nanoparticles as well as larger mineral crystals, and as such can be used to understand both the shape of certain gemstones or crystals with twins.as well as in other areas such as how nanoparticles play a role in the commercial production of chemicals using heterogeneous catalysts.

[1] They include cases for solid particle on substrates, those with internal boundaries and also when growth is important.

An important case is for heterogeneous catalysis where typically the surface of metal nanoparticles is where chemical reactions are taking place.

To optimize the reactions a large metal surface area is desirable, but for stability the nanoparticles need to be supported on a substrate.

These can occur either by accident (growth twins), or can be an integral part of the structure as in decahedral or icosahedral particles.

To understand the shape of particles with twin boundaries a modified Wulff construction is used.

There are related constructions which have been proposed for other cases such as with alloying or when the interface between a nanoparticle and substrate is not flat.

It has the form that the perpendicular distance from a common center to all the external facets is proportional to the surface free energy of each one.

A common approach is to construct the planes normal to the vectors from the center to the surface free energy curve, with the Wulff shape the inner envelope.

This is represented in the Wulff construction figure where the surface free energy is in red, and the single crystal shape would be in blue.

[4][5] For the extended constructions, one or more additional terms are included for interface free energies, for instance the

Comparable cases are generated when the surface free energy is replaced by a growth velocity, these applying for kinetic shapes.

[9][10] If the energy for the interface is very high then the particle has the same shape as it would have in isolation, and effectively dewets the substrate.

[12][13][14] A related form has also been used for precipitates at boundaries, with semi-Wulff construction shapes on both sides.

[15][16] This form was proposed as an extension of the Winterbottom construction (and a play on words) by Jean Taylor.

They often have re-entrant surfaces at the twin boundaries, a phenomenon reported in the 19th century and described in encyclopedias of crystal shapes.

[26] There can also be three twin boundaries per segment where twenty units assemble to form an icosahedral structure.

[32][33] The approach to model these is similar to the Winterbottom construction, now adding an extra facet of energy per unit area half that of the twin boundary -- half so the energy per unit area of the two adjacent segments sums to a full twin boundary energy, and the facets that for the twin boundary are identical for thee segments.

It is named after Georg Wulff, but his paper[1] was not in fact on thermodynamics, but rather on growth kinetics.

This means that fast-growing facets are often not present, for instance often {100} for a face-centered cubic material; the external shape may be dominated by the slowest-growing faces.

[41][2] There are analogues of all the earlier cases when kinetic control dominates:[46][2] There can also be faster growth at re-entrant surfaces around twin boundaries,[47] at the interface for a Winterbottom case, at dislocations[48] and possibly at disclinations, all of which can lead to different shapes.

[49] For instance, faster growth at twin boundaries leads to regular polyhedra such as pentagonal bipyramids for the fivelings with sharp corners and edges, and sharp icosahedral for the particles made of twenty subunits.

In particular: These variants of the Wulff construction correlate well to many shapes found experimentally, but do not explain everything.

Sometimes the shapes with multiple different units are due to coalescence, sometimes they are less symmetric and sometimes, as in Janus particles (for the two-headed god) they contain two materials as illustrated in the figure.

[39] For instance during growth at elevated temperatures a neck can form between two particles, and they can start to merge.

The support can also play a role in reducing sintering by stabilizing the particles so there is less reduction in their surface area with extended use -- larger particles produced by sintering small ones have less surface area for the same total number of atoms.

[71] In addition, the substrate can determine the orientation of the nanoparticles, and combined with what surfaces are exposed in the Winterbottom construction there can be different reactivities which has been exploited for prototype catalysts.

[72][73][74][75] As alluded to earlier, many minerals have crystal twins, and these approaches provide methods to explain the morphologies for either kinetic or thermodynamic control for shapes found in the literature[19] for in marcasite,[76][77] and by Gustav Rose in 1831 for gold.

[70] The decahedral particles and, to a lesser extent the icosahedral ones have shapes determined by the modified Wulff construction.

[2] Note that due to the discrete nature of atoms there can be deviations from the continuum shapes at very small sizes.