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 twinned nanoparticles being explored for other applications such as drug delivery or optical communications. They are also relevant for macroscopic crystals with twins.

The simplest forms of these constructions yield the lowest free energy (thermodynamic) shape, or the stable form for a growing isolated 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 facets.

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 twins in the crystals. 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

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 Georg Wulff, but his paper was not in fact on thermodynamics, rather on the growth kinetics. In many cases growth occurs via the nucleation of small islands on the surface then their sideways growth, what is called step-flow growth. The variant where this type growth dominates is the Kinetic Wulff construction.

In the kinetic case 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 when surface diffusion will change the shape towards the equilibrium shape. Most of the shapes in larger mineral crystals are a consequence of kinetic control. The growth rate of different surfaces depends strongly upon the presence of adsorbates, so can vary substantially.

There can also be faster growth at re-entrant surfaces around twin boundaries (see Modified Wulff construction), at screw dislocations and possibly at disclinations.

For completeness, there is a different type of kinetic control of shapes called diffusion control. This leads to more complex shapes such as dendrites.

Winterbottom construction

The Winterbottom construction, named after Walter L. Winterbottom, 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, and leads to a shape which looks similar to that of a truncated single particle, and can resemble that of a liquid drop on a surface. As indicated in the Figure, for very strong bonding between the Nanoparticle and substrate the particle is flat -- it effectively wets the substrate. In contrast, with low bonding there is much less contact, effectively solid dewetting.

The configuration found depends upon the orientation of the substrate, that of the particle as well as the relative orientation of the two. It is not uncommon to have more than one population of particle on substrate configurations in practice. There is also some dependence upon whether there are steps at the interface.

Summertop construction and friends

This form was proposed as an extension of the Winterbottom construction (and a play on words) by Jean Taylor. It applies to the case of a nanoparticle at a corner. Instead of just using one extra facet for the interface two are included. There are other related extensions, such as solutions with two parallel planes.

Modified Wulff construction

Spinel law contact twinning. A single crystal is shown at left with the composition plane in red. At right, the crystal has effectively been cut on the composition plane and the front half rotated by 180° to produce a contact twin. This creates reentrants at the top, lower left, and lower right of the composition plane.

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. In many cases the twin boundary energy is small compared to an external surface energy, so a single twin is close to half a single crystal rotated by 180 degrees.

There can also be two twin boundaries, which leads to a shape that Cleveland and Uzi Landman called a Marks decahedron when it occurs in face centered cubic materials with five units forming a cyclic twin. Three twin boundaries where twenty units assemble to form an icosahedral structure. Both the decahedral and icosahedral forms can be the most stable ones at the nanoscale.

The above is for the thermodynamic shape. When kinetic growth dominates the velocity of the buried twin boundaries is zero. This can lead to cyclic twins with very sharp shapes.

Caveats

Two materials A and B can be miscible when hot, then phase separate to form Janus particles.

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. There are also some assumptions such as that the substrate remains flat in the Winter bottom construction. This does not have to be the case, the particle can partially or completely be buried by the substrate.