User:Ldm1954/Sandbox/Winterbottom construction

Extended Wulff constructions refer to a number of variants of the base Wulff construction which would be used for a solid single crystal in isolation. They include cases for solid particle on substrates, those with twins and also when growth is important. They are important for many applications such as supported metal nanoparticles used in heterogeneous catalysis or for understanding the shape of nanoparticles being explored for...

These constructions yield the lowest free energy (thermodynamic) shape, or the stable form for a growing particle; it can be difficult to differentiate between the two in experimental data. The thermodynamic cases involve the surface free energy of different facets; the term surface tension refers to liquids, not solids. Those during growth involve the growth velocity of the different surface fscets.

General description

The thermodynamic Wulff construction describes the relationship between the shape of a single crystal and the surface free energy of different surfaces facets. 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. It is named after ...Wulff, but his paper was not in fact on thermodynamics, rather on the growth kinetics. The variant where growth is considered is the #Kinetic Wulff construction.

While the thermodynamic and kinetic constructions are important for free standing particles, often in technological applications particles are on supports. An important case is 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. The problem of the shape on a flat substrate is solved via the #Winterbottom construction.

All the above are for single crystals, but it is common to have crystal twinning. 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.

All these add some additional terms to the base Wulff construction. 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.

Kinetic Wulff construction

This refers to the problem initially considered by Wulff. The distance from the origin to each surface facet is proportional to the growth rate of the facet. This means that fast growing facets are often not present, for instance often {100} for a face centered cubic material, although they will reappear if the crystal is annealed. There can also be faster growth at re-entrant surfaces around twin boundaries (see #Modified Wulff construction), at screw dislocations and possibly at disclinations.

Winterbottom construction

The Winterbottom construction is the solution for the shape of a solid particle on a fixed substrate, where the substrate is forced to remain flat. It is related to the Wulff construction, with the addition of an extra term to describe the interface. These shapes are often found for supported nanoparticles. This leads to a shape which looks similar to that of a truncated single particle, and can resemble that if a liquid drop on a surface.

Summertop construction

This form was proposed as an extension of the Winterbottom construction (and a play on words) by Taylor. It applies to the case of a nanoparticle at a corner.

Modified Wulff construction

In many materials there are twins, which often correspond to a mirroring on a specific plane. For instance, a {111} plane for a face centered material such as gold is the normal twin plane. The construction with twins is somewhat 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. This then leads to re-entrant surfaces at the twin boundaries, a phenomenon reported in the 19th century and described in the encyclopedia of crystal shapes.

There can also be two twin boundaries, which leads to a shape that Cleveland and Landman called a Marks decahedron when it occurs in face centered cubic materials with five units forming a cyclic twin. There can also be three twin boundaries where twenty units assemble to form an icosahedral structure.

Caveats

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.