Extended Wulff constructions refers to a number of different ways to model the structure of nanoparticles as well as larger mineral crystals. They can be used to understand the shape of gemstones and crystals with twins, and in other areas such as understanding both the shape and how nanoparticles play a role in the commercial production of chemicals using heterogeneous catalysts. Extended Wulff constructions are variants of the Wulff construction, which is used for a solid single crystal in isolation. They include cases for solid particles on substrates, those with internal boundaries and also when growth is important.
Depending upon whether there are twins or a substrate, there are different cases as indicated in the decision tree figure. The simplest forms of these constructions yield the lowest Gibbs free energy (thermodynamic) shape, or the stable growth form for an isolated particle; it can be difficult to differentiate between the two in experimental data. The thermodynamic cases involve the surface energy of different facets; the term surface tension refers to liquids, not solids. The shapes found due to growth kinetics involve the growth velocity of the different surface facets. While the thermodynamic and kinetic constructions are relevant for free standing particles, often in technological applications particles are on supports. 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. 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.
General form The thermodynamic Wulff construction describes the relationships between the shape of a single crystal and the surface free energy of different surface 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. This can be viewed as a relationship between the different surface energies and the distance from a Wulff center h j = λ γ j {\displaystyle h_{j}=\lambda \gamma _{j}} , where the vector h j {\displaystyle h_{j}} is the "height" of the j {\displaystyle j} th face, drawn from the center to the face with a surface free energy of γ j {\displaystyle \gamma _{j}} , and λ {\displaystyle \lambda } a scale. 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. In a more mathematical formalism it can be written describing the shape as a set of points S w {\displaystyle S_{w}} given by
S w = x : x . n ^ ≤ λ γ ( n ^ ) {\displaystyle S_{w}=x:x.{\widehat {n}}\leq \lambda \gamma ({\widehat {n}})}
for all unit vectors n ^ {\displaystyle {\widehat {n}}} . For the extended constructions, one or more additional terms are included for interface free energies, for instance the γ i {\displaystyle \gamma _{i}} marked in purple with dashes in the figure. The additional interfaces may be a solid interface for the Winterbottom case, two interfaces for summertop, and one, two, or three twin boundaries for the modified Wulff construction. Comparable cases are generated when the surface free energy is replaced by a growth velocity, these applying for kinetic shapes.
Winterbottom construction
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