The Shapes of Things

Above, $\gamma$ has been assumed to be isotropic. Under this assumption, a finite (isolated) volume body will reduce its total surface energy to a minimum. The result for an isolated body for isotropic surface tension is a sphere.

Figure 35-9: The minimizing surface for a fixed volume with isotropic surface tension.

However, for crystals, $\gamma$ is a function of the orientation of the surface $\gamma(\hat{n})$. For example, in 2-D

Figure: Example of how the surface tension might depend on the orientation of $\hat{n}$ of a surface.

The shape is given by the Wulff construction:

Figure: Example of the Wulff construction to calculate the minimizing surface for a fixed volume with anisotropic surface tension $\gamma(\hat{n})$. The interior envelope in the right figure is the minimizing shape in two dimensions.

The Wulff construction is performed as follows:²

For each orientation $\hat{n}$, draw a ray from the origin to the surface of $\gamma(\hat{n})$. At the end of each ray, construct the perpendicular half plane. The interior of the envelope that results from all such half planes is the minimizing shape for a finite isolated volume.