# Background¶

The equilibrium shape of a droplet of water in a spaceship is a sphere. The reason is that the sphere is the shape that has the smallest area for a given volume, which means that a spherical water droplet minimizes its energy.

The situation is more complicated for crystalline materials, because the surface energy is a function of direction in such materials. Given a direction dependent surface energy $$\gamma[\boldsymbol{n}]$$, the Wulff construction provides the lowest energy shape [Wul01]. The construction can be performed for each direction $$\boldsymbol{n}$$ drawing a plane with $$\boldsymbol{n}$$ at a distance $$\gamma[\boldsymbol{n}]$$ from the origin, and then constructing the inner envelope of those planes. In mathematical terms (and slightly simplified), the construction is thus the set of points

$\mathcal{W} = \left\lbrace \boldsymbol{x}:\, \boldsymbol{x} \cdot \boldsymbol{n} \leq \gamma\left[\boldsymbol{n}\right]\, \text{for all}\, \boldsymbol{n} \right\rbrace.$

The only input needed to make a Wulff construction is thus the direction dependent surface energy $$\gamma[\boldsymbol{n}]$$. It must obey the symmetry of the crystal, which significantly reduces the data needed. In practice, a good approximation to $$\mathcal{W}$$ can usually be obtained by considering only a small number of directions $$\boldsymbol{n}$$ corresponding to low-index facets of the crystal. A Wulff construction can thus often be made based on the input from only a very small number of energy calculations, often based on density functional theory. In WulffPack, a Wulff construction is performed by creating a SingleCrystal object.

## Wulff constructions of decahedra and icosahedra¶

The regular Wulff construction described above assumes a defect-free single crystal. It has been demonstrated many times that small nanoparticles made of FCC metals but with twin boundaries often have a lower energy than single crystalline nanoparticles with the same shape. Two particularly common such particles are based on decahedral and icosahedral symmetry. They can be viewed as consisting of five and twenty tetrahedral grains, respepctively. Fortunately, a Wulff construction can be made for such particles as well [Mar83] (see Decahedron and Icosahedron). WulffPack also allows various aspects of Wulff constructions of these shapes to be compared, see further under Compare shapes.

## Particles on flat surfaces¶

When a particle is placed on a flat surface, symmetry is broken and the equilibrium shapes is changed. A Wulff construction can still be made as long as the broken symmetry is taken into account. Such a construction is usually referred to as a Winterbottom construction [Win67]. In WulffPack, this is handled with a Winterbottom object.